A sundial is a device that tells the time of day when there is sunlight by the apparent position of the Sun in the sky. In the narrowest sense of the word, it consists of a flat plate (the dial) and a gnomon, which casts a shadow onto the dial. As the Sun appears to move across the sky, the shadow aligns with different hour-lines, which are marked on the dial to indicate the time of day. The style is the time-telling edge of the gnomon, though a single point or nodus may be used. The gnomon casts a broad shadow; the shadow of the style shows the time. The gnomon may be a rod, wire, or elaborately decorated metal casting. The style must be parallel to the axis of the Earth's rotation for the sundial to be accurate throughout the year. The style's angle from horizontal is equal to the sundial's geographical latitude.

In a broader sense, a sundial is any device that uses the Sun's altitude or azimuth (or both) to show the time. In addition to their time-telling function, sundials are valued as decorative objects, literary metaphors, and objects of mathematical study.

It is common for inexpensive, mass-produced decorative sundials to have incorrectly aligned gnomons and hour-lines, which cannot be adjusted to tell correct time.[2]

There are several different types of sundials. Some sundials use a shadow or the edge of a shadow while others use a line or spot of light to indicate the time.

The shadow-casting object, known as a gnomon, may be a long thin rod or other object with a sharp tip or a straight edge. Sundials employ many types of gnomon. The gnomon may be fixed or moved according to the season. It may be oriented vertically, horizontally, aligned with the Earth's axis, or oriented in an altogether different direction determined by mathematics.

Given that sundials use light to indicate time, a line of light may be formed by allowing the sun's rays through a thin slit or focusing them through a cylindrical lens. A spot of light may be formed by allowing the sun's rays to pass through a small hole or by reflecting them from a small circular mirror.

Sundials also may use many types of surfaces to receive the light or shadow. Planes are the most common surface, but partial spheres, cylinders, cones and other shapes have been used for greater accuracy or beauty.

Sundials differ in their portability and their need for orientation. The installation of many dials requires knowing the local latitude, the precise vertical direction (e.g., by a level or plumb-bob), and the direction to true North. Portable dials are self-aligning: for example, it may have two dials that operate on different principles, such as a horizontal and analemmatic dial, mounted together on one plate. In these designs, their times agree only when the plate is aligned properly.

Sundials indicate the local solar time, unless corrected for some other time. To obtain the official clock time, three types of corrections need to be made.

Firstly, the orbit of the Earth is not perfectly circular and its rotational axis not perfectly perpendicular to its orbit. The sundial's indicated solar time thus varies from clock time by small amounts that change throughout the year. This correction — which may be as great as 15 minutes — is described by the equation of time. A sophisticated sundial, with a curved style or hour lines, may incorporate this correction. Often instead, simpler sundials are used, with a small plaque that gives the offsets at various times of the year.

Secondly, the solar time must be corrected for the longitude of the sundial relative to the longitude of the official time zone. For example, a sundial located west of Greenwich, England but within the same time-zone, shows an earlier time than the official time. For example, it will show "noon" after the official noon has passed. This correction is often made by rotating the hour-lines by an angle equal to the difference in longitudes.

Lastly, to adjust for daylight saving time, the sundial must shift the time away from solar time by some amount, usually an hour. This correction may be made in the adjustment plaque, or by numbering the hour-lines with two sets of numbers.

Top view of an equatorial sundial. The hour lines are spaced equally about the circle, and the shadow of the gnomon (a thin cylindrical rod) rotates uniformly. The height of the gnomon is 5/12 the outer radius of the dial. This animation depicts the motion of the shadow from 3 a.m. to 9 p.m. (not accounting for Daylight Saving Time) on or around Solstice, when the sun is at its highest declination (roughly 23.5°). Sunrise and sunset occur at 3am and 9pm, respectively, on that day at geographical latitudes near 57.05°, roughly the latitude of Aberdeen, Scotland or Sitka, Alaska.

The principles of sundials are understood most easily from the Sun's apparent motion.[3] The Earth rotates on its axis, and revolves in an elliptical orbit around the Sun. An excellent approximation assumes that the Sun revolves around a stationary Earth on the celestial sphere, which rotates every 24 hours about its celestial axis. The celestial axis is the line connecting the celestial poles. Since the celestial axis is aligned with the axis about which the Earth rotates, the angle of the axis with the local horizontal is the local geographical latitude.

Unlike the fixed stars, the Sun changes its position on the celestial sphere, being at a positive declination in spring and summer, and at a negative declination in autumn and winter, and having exactly zero declination (i.e., being on the celestial equator) at the equinoxes. The Sun's celestial longitude also varies, changing by one complete revolution per year. The path of the Sun on the celestial sphere is called the ecliptic. The ecliptic passes through the twelve constellations of the zodiac in the course of a year.

This model of the Sun's motion helps to understand sundials. If the shadow-casting gnomon is aligned with the celestial poles, its shadow will revolve at a constant rate, and this rotation will not change with the seasons. This is the most common design. In such cases, the same hour lines may be used throughout the year. The hour-lines will be spaced uniformly if the surface receiving the shadow is either perpendicular (as in the equatorial sundial) or circular about the gnomon (as in the armillary sphere).

In other cases, the hour-lines are not spaced evenly, even though the shadow rotates uniformly. If the gnomon is not aligned with the celestial poles, even its shadow will not rotate uniformly, and the hour lines must be corrected accordingly. The rays of light that graze the tip of a gnomon, or which pass through a small hole, or reflect from a small mirror, trace out a cone aligned with the celestial poles. The corresponding light-spot or shadow-tip, if it falls onto a flat surface, will trace out a conic section, such as a hyperbola, ellipse or (at the North or South Poles) a circle.

This conic section is the intersection of the cone of light rays with the flat surface. This cone and its conic section change with the seasons, as the Sun's declination changes; hence, sundials that follow the motion of such light-spots or shadow-tips often have different hour-lines for different times of the year. This is seen in shepherd's dials, sundial rings, and vertical gnomons such as obelisks. Alternatively, sundials may change the angle or position (or both) of the gnomon relative to the hour lines, as in the analemmatic dial or the Lambert dial.

The earliest sundials known from the archaeological record are shadow clocks (1500 BC) from ancient Egyptian astronomy and Babylonian astronomy. Presumably, humans were telling time from shadow-lengths at an even earlier date, but this is hard to verify. In roughly 700 BC, the Old Testament describes a sundial — the "dial of Ahaz" mentioned in Isaiah 38:8 and 2 Kings 20:11. The Roman writer Vitruvius lists dials and shadow clocks known at that time. A canonical sundial is one that indicates the canonical hours of liturgical acts. Such sundials were used from the 7th to the 14th centuries by the members of religious communities. Italian astronomer Giovanni Padovani published a treatise on the sundial in 1570, in which he included instructions for the manufacture and laying out of mural (vertical) and horizontal sundials. Giuseppe Biancani's Constructio instrumenti ad horologia solaria (ca. 1620) discusses how to make a perfect sundial. They have been commonly used since the 16th century.

A London type horizontal dial. The western edge of the gnomon is used as the style before noon, the eastern edge after that time. The changeover causes a discontinuity, the noon gap, in the time scale.

In general, sundials indicate the time by casting a shadow or throwing light onto a surface known as a dial face or dial plate. Although usually a flat plane, the dial face may also be the inner or outer surface of a sphere, cylinder, cone, helix, and various other shapes.

The time is indicated where a shadow or light falls on the dial face, which is usually inscribed with hour lines. Although usually straight, these hour lines may also be curved, depending on the design of the sundial (see below). In some designs, it is possible to determine the date of the year, or it may be required to know the date to find the correct time. In such cases, there may be multiple sets of hour lines for different months, or there may be mechanisms for setting/calculating the month. In addition to the hour lines, the dial face may offer other data—such as the horizon, the equator and the tropics—which are referred to collectively as the dial furniture.

The entire object that casts a shadow or light onto the dial face is known as the sundial's gnomon.[4] However, it is usually only an edge of the gnomon (or another linear feature) that casts the shadow used to determine the time; this linear feature is known as the sundial's style. The style is usually aligned parallel to the axis of the celestial sphere, and therefore is aligned with the local geographical meridian. In some sundial designs, only a point-like feature, such as the tip of the style, is used to determine the time and date; this point-like feature is known as the sundial's nodus.[4][a] Some sundials use both a style and a nodus to determine the time and date.

The gnomon is usually fixed relative to the dial face, but not always; in some designs such as the analemmatic sundial, the style is moved according to the month. If the style is fixed, the line on the dial plate perpendicularly beneath the style is called the substyle,[4] meaning "below the style". The angle the style makes with the plane of the dial plate is called the substyle height, an unusual use of the word height to mean an angle. On many wall dials, the substyle is not the same as the noon line (see below). The angle on the dial plate between the noon line and the substyle is called the substyle distance, an unusual use of the word distance to mean an angle.

By tradition, many sundials have a motto. The motto is usually in the form of an epigram: sometimes sombre reflections on the passing of time and the brevity of life, but equally often humorous witticisms of the dial maker.[5][6]

A dial is said to be equiangular if its hour-lines are straight and spaced equally. Most equiangular sundials have a fixed gnomon style aligned with the Earth's rotational axis, as well as a shadow-receiving surface that is symmetrical about that axis; examples include the equatorial dial, the equatorial bow, the armillary sphere, the cylindrical dial and the conical dial. However, other designs are equiangular, such as the Lambert dial, a version of the analemmatic dial with a moveable style.

Southern-hemisphere sundial in Perth, Australia. Magnify to see that the hour marks run anticlockwise. Note graph of Equation of Time, needed to correct sundial readings.

A sundial at a particular latitude in one hemisphere must be reversed for use at the opposite latitude in the other hemisphere.[7] A vertical direct south sundial in the Northern Hemisphere becomes a vertical direct north sundial in the Southern Hemisphere. To position a horizontal sundial correctly, one has to find true North or South. The same process can be used to do both.[8] The gnomon, set to the correct latitude, has to point to the true South in the Southern hemisphere as in the Northern Hemisphere it has to point to the true North. [9] The hour numbers also run in opposite directions, so on a horizontal dial they run anticlockwise (US: counterclockwise) rather than clockwise.[10]

Sundials which are designed to be used with their plates horizontal in one hemisphere can be used with their plates vertical at the complementary latitude in the other hemisphere. For example, the illustrated sundial in Perth, Australia, which is at latitude 32 degrees South, would function properly if it were mounted on a south-facing vertical wall at latitude 58 (i.e. 90–32) degrees North, which is slightly further North than Perth, Scotland. The surface of the wall in Scotland would be parallel with the horizontal ground in Australia (ignoring the difference of longitude), so the sundial would work identically on both surfaces. Correspondingly, the hour marks, which run counterclockwise on a horizontal sundial in the southern hemisphere, also do so on a vertical sundial in the northern hemisphere. (See the first two illustrations at the top of this article.) On horizontal northern-hemisphere sundials, and on vertical southern-hemisphere ones, the hour marks run clockwise.

The most common reason for a sundial to differ greatly from clock time is that the sundial has not been oriented correctly or its hour lines have not been drawn correctly. For example, most commercial sundials are designed as horizontal sundials as described above. To be accurate, such a sundial must have been designed for the local geographical latitude and its style must be parallel to the Earth's rotational axis; the style must be aligned with true North and its height (its angle with the horizontal) must equal the local latitude. To adjust the style height, the sundial can often be tilted slightly "up" or "down" while maintaining the style's north-south alignment.[11]

A standard time zone covers roughly 15° of longitude, so any point within that zone which is not on the reference longitude (generally a multiple of 15°) will experience a difference from standard time equal to 4 minutes of time per degree. For illustration, sunsets and sunrises are at a much later "official" time at the western edge of a time-zone, compared to sunrise and sunset times at the eastern edge. If a sundial is located at, say, a longitude 5° west of the reference longitude, its time will read 20 minutes slow, since the sun appears to revolve around the Earth at 15° per hour. This is a constant correction throughout the year. For equiangular dials such as equatorial, spherical or Lambert dials, this correction can be made by rotating the dial surface by an angle equaling the difference in longitude, without changing the gnomon position or orientation. However, this method does not work for other dials, such as a horizontal dial; the correction must be applied by the viewer.

At its most extreme, time zones can cause official noon, including daylight savings, to occur up to three hours early (the sun is actually on the meridian at official clock time of 3 pm). This occurs in the far west of Alaska, China, and Spain. For more details and examples, see Skewing of time zones.

The Whitehurst & Son sundial made in 1812, with a circular scale showing the equation of time correction. This is now on display in the Derby Museum.

Although the Sun appears to rotate nearly uniformly about the Earth, it is not perfectly uniform. This is due to the ellipticity of the Earth's orbit (the fact that the Earth's orbit about the Sun is not perfectly circular) and the tilt (obliquity) of the Earth's rotational axis relative to the plane of its orbit. Therefore, sundial time varies from standard clock time. On four days of the year, the correction is effectively zero. However, on others, it can be as much as a quarter-hour early or late. The amount of correction is described by the equation of time. This correction is universal; it does not depend on the local latitude of the sundial. It does, however, change over long periods of time, centuries or more,[12] because of slow variations in the Earth's orbital and rotational motions. Therefore, tables and graphs of the equation of time that were made centuries ago are now significantly incorrect. The reading of an old sundial should be corrected by applying the present-day equation of time, not one from the period when the dial was made.

In some sundials, the equation of time correction is provided as a plaque affixed to the sundial. In more sophisticated sundials, however, the equation can be incorporated automatically. For example, some equatorial bow sundials are supplied with a small wheel that sets the time of year; this wheel in turn rotates the equatorial bow, offsetting its time measurement. In other cases, the hour lines may be curved, or the equatorial bow may be shaped like a vase, which exploits the changing altitude of the sun over the year to effect the proper offset in time.[13]

Sunquest sundial, designed by Richard L. Schmoyer, at the Mount Cuba Observatory in Greenville, Delaware

The Sunquest sundial, designed by Richard L. Schmoyer in the 1950s, uses an analemmic inspired gnomon to cast a shaft of light onto an equatorial time-scale crescent. Sunquest is adjustable for latitude and longitude, automatically correcting for the equation of time, rendering it "as accurate as most pocket watches".[15][16][17][18] Similarly, in place of the shadow of a gnomon the sundial at Miguel Hernández University uses the solar projection of a graph of the equation of time intersecting a time scale to display clock time directly.

Sundial on the Elche Campus of Miguel Hernández University, Spain, which uses a projected graph of the equation of time within the shadow to indicate clock time.

An analemma may be added to many types of sundials to correct apparent solar time to mean solar time or another standard time. These usually have hour lines shaped like "figure eights" (analemmas) according to the equation of time. This compensates for the slight eccentricity in the Earth's orbit and the tilt of the Earth's axis that causes up to a 15-minute variation from mean solar time. This is a type of dial furniture seen on more complicated horizontal and vertical dials.

Prior to the invention of accurate clocks, in the mid-17th Century, sundials were the only timepieces in common use, and were considered to tell the "right" time. The Equation of Time was not used. After the invention of good clocks, sundials were still considered to be correct, and clocks usually incorrect. The Equation of Time was used in the opposite direction from today, to apply a correction to the time shown by a clock to make it agree with sundial time. Some elaborate "equation clocks", such as one made by Joseph Williamson in 1720, incorporated mechanisms to do this correction automatically. (Williamson's clock may have been the first-ever device to use a differential gear.) Only after about 1800 was uncorrected clock time considered to be "right", and sundial time usually "wrong", so the Equation of Time became used as it is today.[citation needed]

The most commonly observed sundials are those in which the shadow-casting style is fixed in position and aligned with the Earth's rotational axis, being oriented with true North and South, and making an angle with the horizontal equal to the geographical latitude. This axis is aligned with the celestial poles, which is closely, but not perfectly, aligned with the (present) pole starPolaris. For illustration, the celestial axis points vertically at the true North Pole, where it points horizontally on the equator. At Jaipur, home of the world's largest sundial, gnomons are raised 26°55" above horizontal, reflecting the local latitude.[20]

On any given day, the Sun appears to rotate uniformly about this axis, at about 15° per hour, making a full circuit (360°) in 24 hours. A linear gnomon aligned with this axis will cast a sheet of shadow (a half-plane) that, falling opposite to the Sun, likewise rotates about the celestial axis at 15° per hour. The shadow is seen by falling on a receiving surface that is usually flat, but which may be spherical, cylindrical, conical or of other shapes. If the shadow falls on a surface that is symmetrical about the celestial axis (as in an armillary sphere, or an equatorial dial), the surface-shadow likewise moves uniformly; the hour-lines on the sundial are equally spaced. However, if the receiving surface is not symmetrical (as in most horizontal sundials), the surface shadow generally moves non-uniformly and the hour-lines are not equally spaced; one exception is the Lambert dial described below.

Some types of sundials are designed with a fixed gnomon that is not aligned with the celestial poles like a vertical obelisk. Such sundials are covered below under the section, "Nodus-based sundials".

The formulas shown in the paragraphs below allow the positions of the hour-lines to be calculated for various types of sundial. In some cases, the calculations are simple; in others they are extremely complicated. There is an alternative, simple method of finding the positions of the hour-lines which can be used for many types of sundial, and saves a lot of work in cases where the calculations are complex.[21] This is an empirical procedure in which the position of the shadow of the gnomon of a real sundial is marked at hourly intervals. The equation of time must be taken into account to ensure that the positions of the hour-lines are independent of the time of year when they are marked. An easy way to do this is to set a clock or watch so it shows "sundial time"[b] which is standard time,[c] plus the equation of time on the day in question.[d][22] The hour-lines on the sundial are marked to show the positions of the shadow of the style when this clock shows whole numbers of hours, and are labelled with these numbers of hours. For example, when the clock reads 5:00, the shadow of the style is marked, and labelled "5" (or "V" in Roman numerals). If the hour-lines are not all marked in a single day, the clock must be adjusted every day or two to take account of the variation of the equation of time.

The distinguishing characteristic of the equatorial dial (also called the equinoctial dial) is the planar surface that receives the shadow, which is exactly perpendicular to the gnomon's style.[23][24][25] This plane is called equatorial, because it is parallel to the equator of the Earth and of the celestial sphere. If the gnomon is fixed and aligned with the Earth's rotational axis, the sun's apparent rotation about the Earth casts a uniformly rotating sheet of shadow from the gnomon; this produces a uniformly rotating line of shadow on the equatorial plane. Since the sun rotates 360° in 24 hours, the hour-lines on an equatorial dial are all spaced 15° apart (360/24).

HE=15∘×t(hours){\displaystyle H_{E}=15^{\circ }\times t(hours)}

The uniformity of their spacing makes this type of sundial easy to construct. If the dial plate material is opaque, both sides of the equatorial dial must be marked, since the shadow will be cast from below in winter and from above in summer. With translucent dial plates (e.g. glass) the hour angles need only be marked on the sun-facing side, although the hour numberings (if used) need be made on both sides of the dial, owing to the differing hour schema on the sun-facing and sun-backing sides. Another major advantage of this dial is that equation of time (EoT) and daylight saving time (DST) corrections can be made by simply rotating the dial plate by the appropriate angle each day. This is because the hour angles are equally spaced around the dial. For this reason, an equatorial dial is often a useful choice when the dial is for public display and it is desirable to have it show the true local time to reasonable accuracy. The EoT correction is made via the relation :

Near the equinoxes in spring and autumn, the sun moves on a circle that is nearly the same as the equatorial plane; hence, no clear shadow is produced on the equatorial dial at those times of year, a drawback of the design.

A nodus is sometimes added to equatorial sundials, which allows the sundial to tell the time of year. On any given day, the shadow of the nodus moves on a circle on the equatorial plane, and the radius of the circle measures the declination of the sun. The ends of the gnomon bar may be used as the nodus, or some feature along its length. An ancient variant of the equatorial sundial has only a nodus (no style) and the concentric circular hour-lines are arranged to resemble a spider-web.[26]

In the horizontal sundial (also called a garden sundial), the plane that receives the shadow is aligned horizontally, rather than being perpendicular to the style as in the equatorial dial.[27][28][29] Hence, the line of shadow does not rotate uniformly on the dial face; rather, the hour lines are spaced according to the rule [30][31]

where L is the sundial's geographical latitude (and the angle the gnomon makes with the dial plate), HH{\displaystyle H_{H}} is the angle between a given hour-line and the noon hour-line (which always points towards true North) on the plane, and t is the number of hours before or after noon. For example, the angle HH{\displaystyle H_{H}} of the 3pm hour-line would equal the arctangent of sin L, since tan 45° = 1. When L equals 90° (at the North Pole), the horizontal sundial becomes an equatorial sundial; the style points straight up (vertically), and the horizontal plane is aligned with the equatorial plane; the hour-line formula becomes HH{\displaystyle H_{H}} = 15° × t, as for an equatorial dial. A horizontal sundial at the Earth's equator, where L equals 0°, would require a (raised) horizontal style and would be an example of a polar sundial (see below).

The chief advantages of the horizontal sundial are that it is easy to read, and the sun lights the face throughout the year. All the hour-lines intersect at the point where the gnomon's style crosses the horizontal plane. Since the style is aligned with the Earth's rotational axis, the style points true North and its angle with the horizontal equals the sundial's geographical latitude L. A sundial designed for one latitude can be adjusted for use at another latitude by tilting its base upwards or downwards by an angle equal to the difference in latitude. For example, a sundial designed for a latitude of 40° can be used at a latitude of 45°, if the sundial plane is tilted upwards by 5°, thus aligning the style with the Earth's rotational axis.[citation needed] Many ornamental sundials are designed to be used at 45 degrees north. Some mass-produced garden sundials fail to correctly calculate the hourlines and so can never be corrected. A local standard time zone is nominally 15 degrees wide, but may be modified to follow geographic or political boundaries. A sundial can be rotated around its style (which must remain pointed at the celestial pole) to adjust to the local time zone. In most cases, a rotation in the range of 7.5 degrees east to 23 degrees west suffices. This will introduce error in sundials that do not have equal hour angles. To correct for daylight saving time, a face needs two sets of numerals or a correction table. An informal standard is to have numerals in hot colors for summer, and in cool colors for winter.[citation needed] Since the hour angles are not evenly spaced, the equation of time corrections cannot be made via rotating the dial plate about the gnomon axis. These types of dials usually have an equation of time correction tabulation engraved on their pedestals or close by. Horizontal dials are commonly seen in gardens, churchyards and in public areas.

In the common vertical dial, the shadow-receiving plane is aligned vertically; as usual, the gnomon's style is aligned with the Earth's axis of rotation.[23][32][33] As in the horizontal dial, the line of shadow does not move uniformly on the face; the sundial is not equiangular. If the face of the vertical dial points directly south, the angle of the hour-lines is instead described by the formula[34][35]

where L is the sundial's geographical latitude, HV{\displaystyle H_{V}} is the angle between a given hour-line and the noon hour-line (which always points due north) on the plane, and t is the number of hours before or after noon. For example, the angle HV{\displaystyle H_{V}} of the 3pm hour-line would equal the arctangent of cos L, since tan 45° = 1. Interestingly, the shadow moves counter-clockwise on a South-facing vertical dial, whereas it runs clockwise on horizontal and equatorial north-facing dials.

Dials with faces perpendicular to the ground and which face directly South, North, East, or West are called vertical direct dials.[36][37] It is widely believed, and stated in respectable publications, that a vertical dial cannot receive more than twelve hours of sunlight a day, no matter how many hours of daylight there are.[38] However, there is an exception. Vertical sundials in the tropics which face the nearer pole (e.g. north facing in the zone between the Equator and the Tropic of Cancer) can actually receive sunlight for more than 12 hours from sunrise to sunset for a short period around the time of the summer solstice. For example, at latitude 20 degrees North, on June 21, the sun shines on a north-facing vertical wall for 13 hours, 21 minutes.[39] Vertical sundials which do not face directly South (in the northern hemisphere) may receive significantly less than twelve hours of sunlight per day, depending on the direction they do face, and on the time of year. For example, a vertical dial that faces due East can tell time only in the morning hours; in the afternoon, the sun does not shine on its face. Vertical dials that face due East or West are polar dials, which will be described below. Vertical dials that face North are uncommon, because they tell time only during the spring and summer, and do not show the midday hours except in tropical latitudes (and even there, only around midsummer). For non-direct vertical dials — those that face in non-cardinal directions — the mathematics of arranging the style and the hour-lines becomes more complicated; it may be easier to mark the hour lines by observation, but the placement of the style, at least, must be calculated first; such dials are said to be declining dials.[40][41][42]

Vertical dials are commonly mounted on the walls of buildings, such as town-halls, cupolas and church-towers, where they are easy to see from far away. In some cases, vertical dials are placed on all four sides of a rectangular tower, providing the time throughout the day. The face may be painted on the wall, or displayed in inlaid stone; the gnomon is often a single metal bar, or a tripod of metal bars for rigidity. If the wall of the building faces toward the South, but does not face due South, the gnomon will not lie along the noon line, and the hour lines must be corrected. Since the gnomon's style must be parallel to the Earth's axis, it always "points" true North and its angle with the horizontal will equal the sundial's geographical latitude; on a direct south dial, its angle with the vertical face of the dial will equal the colatitude, or 90° minus the latitude.[43]

In polar dials, the shadow-receiving plane is aligned parallel to the gnomon-style.[44][45][46] Thus, the shadow slides sideways over the surface, moving perpendicularly to itself as the sun rotates about the style. As with the gnomon, the hour-lines are all aligned with the Earth's rotational axis. When the sun's rays are nearly parallel to the plane, the shadow moves very quickly and the hour lines are spaced far apart. The direct East- and West-facing dials are examples of a polar dial. However, the face of a polar dial need not be vertical; it need only be parallel to the gnomon. Thus, a plane inclined at the angle of latitude (relative to horizontal) under the similarly inclined gnomon will be a polar dial. The perpendicular spacing X of the hour-lines in the plane is described by the formula

X=Htan⁡(15∘×t){\displaystyle X=H\tan(15^{\circ }\times t)}

where H is the height of the style above the plane, and t is the time (in hours) before or after the center-time for the polar dial. The center time is the time when the style's shadow falls directly down on the plane; for an East-facing dial, the center time will be 6am, for a West-facing dial, this will be 6pm, and for the inclined dial described above, it will be noon. When t approaches ±6 hours away from the center time, the spacing X diverges to +∞; this occurs when the sun's rays become parallel to the plane.

Effect of declining on a sundial's hour-lines. A vertical dial, at a latitude of 51° N, designed to face due South (far left) shows all the hours from 6am to 6pm, and has converging hour-lines symmetrical about the noon hour-line. By contrast, a West-facing dial (far right) is polar, with parallel hour lines, and shows only hours after noon. At the intermediate orientations of South-Southwest, Southwest, and West-Southwest, the hour lines are asymmetrical about noon, with the morning hour-lines ever more widely spaced.

Two sundials, a large and a small one, at Fatih Mosque, Istanbul dating back to the late 16th century. It is on the southwest facade with an azimuth angle of 52° N.

A declining dial is any non-horizontal, planar dial that does not face in a cardinal direction, such as (true) North, South, East or West. [40][47][42] As usual, the gnomon's style is aligned with the Earth's rotational axis, but the hour-lines are not symmetrical about the noon hour-line. For a vertical dial, the angle HVD{\displaystyle H_{\text{VD}}} between the noon hour-line and another hour-line is given by the formula below. Note that HVD{\displaystyle H_{\text{VD}}} is defined positive in the clockwise sense w.r.t. the upper vertical hour angle; and that its conversion to the equivalent solar hour requires careful consideration of which quadrant of the sundial that it belongs in.[48]

where L{\displaystyle L} is the sundial's geographical latitude; t is the time before or after noon; D{\displaystyle D} is the angle of declination from true south, defined as positive when east of south; and so{\displaystyle s_{o}} is a switch integer for the dial orientation. A partly south-facing dial has an so{\displaystyle s_{o}} value of + 1; those partly north-facing, a value of -1. When such a dial faces South (D=0∘{\displaystyle D=0^{\circ }}), this formula reduces to the formula given above for vertical south-facing dials, i.e.

When a sundial is not aligned with a cardinal direction, the substyle of its gnomon is not aligned with the noon hour-line. The angle B{\displaystyle B} between the substyle and the noon hour-line is given by the formula[48]

tan⁡B=sin⁡Dcot⁡L{\displaystyle \tan B=\sin D\cot L}

If a vertical sundial faces true South or North (D=0∘{\displaystyle D=0^{\circ }} or D=180∘{\displaystyle D=180^{\circ }}, respectively), the angle B=0∘{\displaystyle B=0^{\circ }} and the substyle is aligned with the noon hour-line.

The height of the gnomon, that is the angle the style makes to the plate, G{\displaystyle G}, is given by :

Vertical reclining dial in the Southern Hemisphere, facing due north, with hyperbolic declination lines and hour lines. Ordinary vertical sundial at this latitude (between tropics) could not produce a declination line for the summer solstice.

The sundials described above have gnomons that are aligned with the Earth's rotational axis and cast their shadow onto a plane. If the plane is neither vertical nor horizontal nor equatorial, the sundial is said to be reclining or inclining.[50] Such a sundial might be located on a South-facing roof, for example. The hour-lines for such a sundial can be calculated by slightly correcting the horizontal formula above[51]

where R{\displaystyle R} is the desired angle of reclining relative to the local vertical, L is the sundial's geographical latitude, HRV{\displaystyle H_{RV}} is the angle between a given hour-line and the noon hour-line (which always points due north) on the plane, and t is the number of hours before or after noon. For example, the angle HRV{\displaystyle H_{RV}} of the 3pm hour-line would equal the arctangent of cos(L + R), since tan 45° = 1. When R equals 0° (in other words, a South-facing vertical dial), we obtain the vertical dial formula above.

Some authors use a more specific nomenclature to describe the orientation of the shadow-receiving plane. If the plane's face points downwards towards the ground, it is said to be proclining or inclining, whereas a dial is said to be reclining when the dial face is pointing away from the ground. Many authors also often refer to reclined, proclined and inclined sundials in general as inclined sundials. It is also common in the latter case to measure the angle of inclination relative to the horizontal plane on the sun side of the dial. In such texts, since I = 90° + R, the hour angle formula will often be seen written as :

Some sundials both decline and recline, in that their shadow-receiving plane is not oriented with a cardinal direction (such as true North or true South) and is neither horizontal nor vertical nor equatorial. For example, such a sundial might be found on a roof that was not oriented in a cardinal direction.

The formulae describing the spacing of the hour-lines on such dials are rather more complicated than those for simpler dials.

There are various solution approaches, including some using the methods of rotation matrices, and some making a 3D model of the reclined-declined plane and its vertical declined counterpart plane, extracting the geometrical relationships between the hour angle components on both these planes and then reducing the trigonometric algebra.[52][53]

One system of formulas for Reclining-Declining sundials: (as stated by Fennewick)[54]

The angle HRD{\displaystyle H_{\text{RD}}} between the noon hour-line and another hour-line is given by the formula below. Note that HRD{\displaystyle H_{\text{RD}}} advances counterclockwise with respect to the zero hour angle for those dials that are partly south-facing and clockwise for those that are north-facing.

Here L{\displaystyle L} is the sundial's geographical latitude; so{\displaystyle s_{o}} is the orientation switch integer; t is the time in hours before or after noon; and R{\displaystyle R} and D{\displaystyle D} are the angles of reclination and declination, respectively. Note that R{\displaystyle R} is measured with reference to the vertical. It is positive when the dial leans back towards the horizon behind the dial and negative when the dial leans forward to the horizon on the sun's side. Declination angle D{\displaystyle D} is defined as positive when moving east of true south. Dials facing fully or partly south have so{\displaystyle s_{o}} = +1, while those partly or fully north-facing have an so{\displaystyle s_{o}} value of -1. Since the above expression gives the hour angle as an arctan function, due consideration must be given to which quadrant of the sundial each hour belongs to before assigning the correct hour angle.

Unlike the simpler vertical declining sundial, this type of dial does not always show hour angles on its sunside face for all declinations between east and west. When a northern hemisphere partly south-facing dial reclines back (i.e. away from the sun) from the vertical, the gnomon will become co-planar with the dial plate at declinations less than due east or due west. Likewise for southern hemisphere dials that are partly north-facing. Were these dials reclining forward, the range of declination would actually exceed due east and due west. In a similar way, northern hemisphere dials that are partly north-facing and southern hemisphere dials that are south-facing, and which lean forward toward their upward pointing gnomons, will have a similar restriction on the range of declination that is possible for a given reclination value. The critical declination Dc{\displaystyle D_{c}} is a geometrical constraint which depends on the value of both the dial's reclination and its latitude :

As with the vertical declined dial, the gnomon's substyle is not aligned with the noon hour-line. The general formula for the angle B{\displaystyle B}, between the substyle and the noon-line is given by :

Because of the complexity of the above calculations, using them for the practical purpose of designing a dial of this type is difficult and prone to error. It has been suggested that it is better to locate the hour lines empirically, marking the positions of the shadow of a style on a real sundial at hourly intervals as shown by a clock.[21] See Empirical hour-line marking, above.

The surface receiving the shadow need not be a plane, but can have any shape, provided that the sundial maker is willing to mark the hour-lines. If the style is aligned with the Earth's rotational axis, a spherical shape is convenient since the hour-lines are equally spaced, as they are on the equatorial dial above; the sundial is equiangular. This is the principle behind the armillary sphere and the equatorial bow sundial.[55][56][57] However, some equiangular sundials — such as the Lambert dial described below — are based on other principles.

In the equatorial bow sundial, the gnomon is a bar, slot or stretched wire parallel to the celestial axis. The face is a semicircle, corresponding to the equator of the sphere, with markings on the inner surface. This pattern, built a couple of meters wide out of temperature-invariant steel invar, was used to keep the trains running on time in France before World War I.[58]

Among the most precise sundials ever made are two equatorial bows constructed of marble found in Yantra mandir.[59][60] This collection of sundials and other astronomical instruments was built by Maharaja Jai Singh II at his then-new capital of Jaipur, India between 1727 and 1733. The larger equatorial bow is called the Samrat Yantra (The Supreme Instrument); standing at 27 meters, its shadow moves visibly at 1 mm per second, or roughly a hand's breadth (6 cm) every minute.

Other non-planar surfaces may be used to receive the shadow of the gnomon.

As an elegant alternative, the style (which could be created by a hole or slit in the circumference) may be located on the circumference of a cylinder or sphere, rather than at its central axis of symmetry.

In that case, the hour lines are again spaced equally, but at twice the usual angle, due to the geometrical inscribed angle theorem. This is the basis of some modern sundials, but it was also used in ancient times; [e]

In another variation of the polar-axis-aligned cylindrical, a cylindrical dial could be rendered as a helical ribbon-like surface, with a thin gnomon located either along its center or at its periphery.

Sundials can be designed with a gnomon that is placed in a different position each day throughout the year. In other words, the position of the gnomon relative to the centre of the hour lines varies. The gnomon need not be aligned with the celestial poles and may even be perfectly vertical (the analemmatic dial). These dials, when combined with fixed-gnomon sundials, allow the user to determine true North with no other aid; the two sundials are correctly aligned if and only if they both show the same time.[citation needed]

Universal ring dial. The dial is suspended from the cord shown in the upper left; the suspension point on the vertical meridian ring can be changed to match the local latitude. The center bar is twisted until a sunray passes through the small hole and falls on the horizontal equatorial ring. See Commons annotations for labels.

A universal equinoctial ring dial (sometimes called a ring dial for brevity, although the term is ambiguous), is a portable version of an armillary sundial,[62] or was inspired by the mariner's astrolabe.[63] It was likely invented by William Oughtred around 1600 and became common throughout Europe.[64]

In its simplest form, the style is a thin slit that allows the sun's rays to fall on the hour-lines of an equatorial ring. As usual, the style is aligned with the Earth's axis; to do this, the user may orient the dial towards true North and suspend the ring dial vertically from the appropriate point on the meridian ring. Such dials may be made self-aligning with the addition of a more complicated central bar, instead of a simple slit-style. These bars are sometimes an addition to a set of Gemma's rings. This bar could pivot about its end points and held a perforated slider that was positioned to the month and day according to a scale scribed on the bar. The time was determined by rotating the bar towards the sun so that the light shining through the hole fell on the equatorial ring. This forced the user to rotate the instrument, which had the effect of aligning the instrument's vertical ring with the meridian.

When not in use, the equatorial and meridian rings can be folded together into a small disk.

In 1610, Edward Wright created the sea ring, which mounted a universal ring dial over a magnetic compass. This permitted mariners to determine the time and magnetic variation in a single step.[65]

Analemmatic sundials are a type of horizontal sundial that has a vertical gnomon and hour markers positioned in an elliptical pattern. There are no hour lines on the dial and the time of day is read on the ellipse. The gnomon is not fixed and must change position daily to accurately indicate time of day. Analemmatic sundials are sometimes designed with a human as the gnomon. Human gnomon analemmatic sundials are not practical at lower latitudes where a human shadow is quite short during the summer months. A 66 inch tall person casts a 4-inch shadow at 27 deg latitude on the summer solstice.[66]

The Foster-Lambert dial is another movable-gnomon sundial.[67] In contrast to the elliptical analemmatic dial, the Lambert dial is circular with evenly spaced hour lines, making it an equiangular sundial, similar to the equatorial, spherical, cylindrical and conical dials described above. The gnomon of a Foster-Lambert dial is neither vertical nor aligned with the Earth's rotational axis; rather, it is tilted northwards by an angle α = 45° - (Φ/2), where Φ is the geographical latitude. Thus, a Foster-Lambert dial located at latitude 40° would have a gnomon tilted away from vertical by 25° in a northerly direction. To read the correct time, the gnomon must also be moved northwards by a distance

Y=Rtan⁡αtan⁡δ{\displaystyle Y=R\tan \alpha \tan \delta \,}

where R is the radius of the Foster-Lambert dial and δ again indicates the Sun's declination for that time of year.

Altitude dials measure the height of the sun in the sky, rather than directly measuring its hour-angle about the Earth's axis. They are not oriented towards true North, but rather towards the sun and generally held vertically. The sun's elevation is indicated by the position of a nodus, either the shadow-tip of a gnomon, or a spot of light.

In altitude dials, the time is read from where the nodus falls on a set of hour-curves that vary with the time of year. Many such altitude-dials' construction is calculation-intensive, as also the case with many azimuth dials. But the capuchin dials (described below) are constructed and used graphically.

Altitude dials' disadvantages:

Since the sun's altitude is the same at times equally spaced about noon (e.g., 9am and 3pm), the user had to know whether it was morning or afternoon. At, say, 3:00 pm, that isn't a problem. But when the dial indicates a time 15 minutes from noon, the user likely won't have a way of distinguishing 11:45 from 12:15.

Additionally, altitude dials are less accurate near noon, because the sun's altitude isn't changing rapidly then.

Many of these dials are portable and simple to use. As is often the case with other sundials, many altitude dials are designed for only one latitude. But the capuchin dial (described below) has a version that's adjustable for latitude.[68]

The book on sundials by Mayall & Mayall describes the Universal Capuchin sundial.

The length of a human shadow (or of any vertical object) can be used to measure the sun's elevation and, thence, the time.[69] The Venerable Bede gave a table for estimating the time from the length of one's shadow in feet, on the assumption that a monk's height is six times the length of his foot. Such shadow lengths will vary with the geographical latitude and with the time of year. For example, the shadow length at noon is short in summer months, and long in winter months.

A shepherd's dial — also known as a shepherds' column dial,[70][71]pillar dial, cylinder dial or chilindre — is a portable cylindrical sundial with a knife-like gnomon that juts out perpendicularly.[72] It is normally dangled from a rope or string so the cylinder is vertical. The gnomon can be twisted to be above a month or day indication on the face of the cylinder. This corrects the sundial for the equation of time. The entire sundial is then twisted on its string so that the gnomon aims toward the sun, while the cylinder remains vertical. The tip of the shadow indicates the time on the cylinder. The hour curves inscribed on the cylinder permit one to read the time. Shepherd's dials are sometimes hollow, so that the gnomon can fold within when not in use.

The cylindrical shepherd's dial can be unrolled into a flat plate. In one simple version,[73] the front and back of the plate each have three columns, corresponding to pairs of months with roughly the same solar declination (June–July, May–August, April–September, March–October, February–November, and January–December). The top of each column has a hole for inserting the shadow-casting gnomon, a peg. Often only two times are marked on the column below, one for noon and the other for mid-morning/mid-afternoon.

Timesticks, clock spear,[70] or shepherds' time stick,[70] are based on the same principles as dials.[70][71] The time stick is carved with eight vertical time scales for a different period of the year, each bearing a time scale calculated according to the relative amount of daylight during the different months of the year. Any reading depends not only on the time of day but also on the latitude and time of year.[71] A peg gnomon is inserted at the top in the appropriate hole or face for the season of the year, and turned to the Sun so that the shadow falls directly down the scale. Its end displays the time.[70]

In a ring dial (also known as an Aquitaine or a perforated ring dial), the ring is hung vertically and oriented sideways towards the sun.[74] A beam of light passes through a small hole in the ring and falls on hour-curves that are inscribed on the inside of the ring. To adjust for the equation of time, the hole is usually on a loose ring within the ring so that the hole can be adjusted to reflect the current month.

Card dials are another form of altitude dial.[75] A card is aligned edge-on with the sun and tilted so that a ray of light passes through an aperture onto a specified spot, thus determining the sun's altitude. A weighted string hangs vertically downwards from a hole in the card, and carries a bead or knot. The position of the bead on the hour-lines of the card gives the time. In more sophisticated versions such as the Capuchin dial, there is only one set of hour-lines, i.e., the hour lines do not vary with the seasons. Instead, the position of the hole from which the weighted string hangs is varied according to the season.

The Capuchin sundials are constructed and used graphically, as opposed the direct hour-angle measurements of horizontal or equatorial dials; or the calculated hour angle lines of some altitude and azimuth dials.

In addition to the ordinary Capuchin dial, there is a universal Capuchin dial, adjustable for latitude.

Another type of sundial follows the motion of a single point of light or shadow, which may be called the nodus. For example, the sundial may follow the sharp tip of a gnomon's shadow, e.g., the shadow-tip of a vertical obelisk (e.g., the Solarium Augusti) or the tip of the horizontal marker in a shepherd's dial. Alternatively, sunlight may be allowed to pass through a small hole or reflected from a small (e.g., coin-sized) circular mirror, forming a small spot of light whose position may be followed. In such cases, the rays of light trace out a cone over the course of a day; when the rays fall on a surface, the path followed is the intersection of the cone with that surface. Most commonly, the receiving surface is a geometrical plane, so that the path of the shadow-tip or light-spot (called declination line) traces out a conic section such as a hyperbola or an ellipse. The collection of hyperbolae was called a pelekonon (axe) by the Greeks, because it resembles a double-bladed ax, narrow in the center (near the noonline) and flaring out at the ends (early morning and late evening hours).

Declination lines at solstices and equinox for sundials, located at different latitudes

There is a simple verification of hyperbolic declination lines on a sundial: the distance from the origin to the equinox line should be equal to harmonic mean of distances from the origin to summer and winter solstice lines.[76]

Nodus-based sundials may use a small hole or mirror to isolate a single ray of light; the former are sometimes called aperture dials. The oldest example is perhaps the antiborean sundial (antiboreum), a spherical nodus-based sundial that faces true North; a ray of sunlight enters from the South through a small hole located at the sphere's pole and falls on the hour and date lines inscribed within the sphere, which resemble lines of longitude and latitude, respectively, on a globe.[77]

Isaac Newton developed a convenient and inexpensive sundial, in which a small mirror is placed on the sill of a south-facing window.[78] The mirror acts like a nodus, casting a single spot of light on the ceiling. Depending on the geographical latitude and time of year, the light-spot follows a conic section, such as the hyperbolae of the pelikonon. If the mirror is parallel to the Earth's equator, and the ceiling is horizontal, then the resulting angles are those of a conventional horizontal sundial. Using the ceiling as a sundial surface exploits unused space, and the dial may be large enough to be very accurate.

Sundials are sometimes combined into multiple dials. If two or more dials that operate on different principles — say, such as an analemmatic dial and a horizontal or vertical dial — are combined, the resulting multiple dial becomes self-aligning, most of the time. Both dials need to output both time and declination. In other words, the direction of true North need not be determined; the dials are oriented correctly when they read the same time and declination. However, the most common forms combine dials are based on the same principle and the analemmatic does not normally output the declination of the sun, thus are not self-aligning.[79]

Diptych sundial in the form of a lute, c. 1612. The gnomons-style is a string stretched between a horizontal and vertical face. This sundial also has a small nodus (a bead on the string) that tells time on the hyperbolic pelikinon, just above the date on the vertical face.

The diptych consisted of two small flat faces, joined by a hinge.[80] Diptychs usually folded into little flat boxes suitable for a pocket. The gnomon was a string between the two faces. When the string was tight, the two faces formed both a vertical and horizontal sundial. These were made of white ivory, inlaid with black lacquer markings. The gnomons were black braided silk, linen or hemp string. With a knot or bead on the string as a nodus, and the correct markings, a diptych (really any sundial large enough) can keep a calendar well-enough to plant crops. A common error describes the diptych dial as self-aligning. This is not correct for diptych dials consisting of a horizontal and vertical dial using a string gnomon between faces, no matter the orientation of the dial faces. Since the string gnomon is continuous, the shadows must meet at the hinge; hence, any orientation of the dial will show the same time on both dials.[81]

A common type of multiple dial has sundials on every face of a Platonic solid (regular polyhedron), usually a cube.[82]

Extremely ornate sundials can be composed in this way, by applying a sundial to every surface of a solid object.

In some cases, the sundials are formed as hollows in a solid object, e.g., a cylindrical hollow aligned with the Earth's rotational axis (in which the edges play the role of styles) or a spherical hollow in the ancient tradition of the hemisphaerium or the antiboreum. (See the History section above.) In some cases, these multiface dials are small enough to sit on a desk, whereas in others, they are large stone monuments.

A Polyhedral's dial faces can be designed to give the time for different time-zones simultaneously.

Prismatic dials are a special case of polar dials, in which the sharp edges of a prism of a concave polygon serve as the styles and the sides of the prism receive the shadow.[83] Examples include a three-dimensional cross or star of David on gravestones.

The Benoy Dial was invented by Walter Gordon Benoy of Collingham in Nottinghamshire. Light is used to replace the shadow-edge of a gnomon. Whereas a gnomon casts a sheet of shadow, an equivalent sheet of light can be created by allowing the Sun's rays through a thin slit, reflecting them from a long, slim mirror (usually half-cylindrical), or focusing them through a cylindrical lens. For illustration, the Benoy Dial uses a cylindrical lens to create a sheet of light, which falls as a line on the dial surface. Benoy dials can be seen throughout Great Britain, such as[84]

Invented by the German mathematician Hugo Michnik in 1922, the bifilar sundial has two non-intersecting threads parallel to the dial. Usually the second thread is orthogonal to the first.[86] The intersection of the two threads' shadows gives the local solar time.

A digital sundial indicates the current time with numerals formed by the sunlight striking it. Sundials of this type are installed in the Deutsches Museum in Munich and in the Sundial Park in Genk (Belgium), and a small version is available commercially. There is a patent for this type of sundial.[87]

The globe dial is a sphere aligned with the Earth's rotational axis, and equipped with a spherical vane.[88] Similar to sundials with a fixed axial style, a globe dial determines the time from the Sun's azimuthal angle in its apparent rotation about the earth. This angle can be determined by rotating the vane to give the smallest shadow.

Noon mark from the Greenwich Royal Observatory. The analemma is the narrow figure-8 shape, which plots the equation of time (in degrees, not time, 1°=4 minutes) versus the altitude of the sun at noon at the sundial's location. The altitude is measured vertically, the equation of time horizontally.

The simplest sundials do not give the hours, but rather note the exact moment of 12:00 noon. [89] In centuries past, such dials were used to correct mechanical clocks, which were sometimes so inaccurate as to lose or gain significant time in a single day.

In some U.S. colonial-era houses, a noon-mark can often be found carved into a floor or windowsill.[90] Such marks indicate local noon, and provide a simple and accurate time reference for households that do not possess accurate clocks. In modern times, in some Asian countries, post offices set their clocks from a precision noon-mark. These in turn provide the times for the rest of the society. The typical noon-mark sundial was a lens set above an analemmatic plate. The plate has an engraved figure-eight shape, which corresponds to plotting the equation of time (described above) versus the solar declination. When the edge of the sun's image touches the part of the shape for the current month, this indicates that it is 12:00 noon.

A sundial cannon, sometimes called a 'meridian cannon', is a specialized sundial that is designed to create an 'audible noonmark', by automatically igniting a quantity of gunpowder at noon. These were novelties rather than precision sundials, sometimes installed in parks in Europe mainly in the late 18th or early 19th century. They typically consist of a horizontal sundial, which has in addition to a gnomon a suitably mounted lens, set to focus the rays of the sun at exactly noon on the firing pan of a miniature cannon loaded with gunpowder (but no ball). To function properly the position and angle of the lens must be adjusted seasonally.[citation needed]

The association of sundials with time has inspired their designers over the centuries to display mottoes as part of the design. Often these cast the device in the role of memento mori, inviting the observer to reflect on the transience of the world and the inevitability of death. "Do not kill time, for it will surely kill thee." Other mottoes are more whimsical: "I count only the sunny hours," and "I am a sundial and I make a botch / of what is done far better by a watch." Collections of sundial mottoes have often been published through the centuries.[citation needed]

If a horizontal-plate sundial is made for the latitude in which it is being used, and if it is mounted with its plate horizontal and its gnomon pointing to the celestial pole that is above the horizon, then it shows the correct time in apparent solar time. Conversely, if the directions of the cardinal points are initially unknown, but the sundial is aligned so it shows the correct apparent solar time as calculated from the reading of a clock, its gnomon shows the direction of True North or South, allowing the sundial to be used as a compass. The sundial can be placed on a horizontal surface, and rotated about a vertical axis until it shows the correct time. The gnomon will then be pointing to the North, in the northern hemisphere, or to the South in the southern hemisphere. This method is much more accurate than using a watch as a compass (see Cardinal direction#Watch face) and can be used in places where the magnetic declination is large, making a magnetic compass unreliable. An alternative method uses two sundials of different designs. (See #Multiple dials, above.) The dials are attached to and aligned with each other, and are oriented so they show the same time. This allows the directions of the cardinal points and the apparent solar time to be determined simultaneously, without requiring a clock.[citation needed]

^In some technical writing, the word "gnomon" can also mean the perpendicular height of a nodus from the dial plate. The point where the style intersects the dial plate is called the gnomon root.

^A clock showing sundial time always agrees with a sundial in the same locality.

^Strictly, local mean time rather than standard time should be used. However, using standard time makes the sundial more useful, since it does not have to be corrected for time zone or longitude.

^The equation of time is considered to be positive when "sundial time" is ahead of "clock time", negative otherwise. See the graph shown in the section #Equation of time correction, above. For example, if the equation of time is -5 minutes and the standard time is 9:40, the sundial time is 9:35.

^Chaucer:as in his Parson's Tale. It was four o'clock according to my guess,
Since eleven feet, a little more or less,
my shadow at the time did fall,
Considering that I myself am six feet tall.

^Henry VI, Part 3:O God! methinks it were a happy life
To be no better than a homely swain;
To sit upon a hill, as I do now,
To carve out dials, quaintly, point by point,
Thereby to see the minutes, how they run--
How many makes the hour full complete,
How many hours brings about the day,
How many days will finish up the year,
How many years a mortal man may live.

^For example, in the Chaucer's Canterbury Tales, the monk says, "Goth now your wey," quod he, "al stille and softe,
And lat us dyne as sone as that ye may;
for by my chilindre it is pryme of day."

^Moss, Tony. "How do sundials work". British Sundial society. Archived from the original on August 2, 2013. Retrieved 21 September 2013. This ugly plastic ‘non-dial’ does nothing at all except display the ‘designer’s ignorance and persuade the general public that ‘real’ sundials don’t work.

Rohr, RRJ (1996). Sundials: History, Theory, and Practice (translated by G. Godin ed.). New York: Dover. ISBN0-486-29139-1. Slightly amended reprint of the 1970 translation published by University of Toronto Press (Toronto). The original was published in 1965 under the title Les Cadrans solaires by Gauthier-Villars (Montrouge, France).

1.
Aldeburgh
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Aldeburgh /ˈɔːlbrə/ is a coastal town in the English county of Suffolk. It remains an artistic and literary centre with an annual Poetry Festival and several festivals as well as other cultural events. It is a former Tudor port and was granted Borough status in 1529 by Henry VIII and its historic buildings include a 16th-century moot hall and a Napoleonic-era Martello Tower. Second homes make up roughly a third of the residential property. The town is a tourist destination with visitors attracted by its Blue Flag shingle beach and fisherman huts, where fish are sold daily. Two family-run fish and chip shops are cited as among the best in the UK, Alde Burgh means old fort although this structure, along with much of the Tudor town, has now been lost to the sea. In the 16th century, Aldeburgh was a port, and had a flourishing ship-building industry. Sir Francis Drakes Greyhound and Pelican were both built in Aldeburgh, the flagship of the Virginia Company, the Sea Venture is believed to have been built here in 1608. Aldeburghs importance as a port declined as the River Alde silted up and it survived mainly as a fishing village until the 19th century, when it also became a seaside resort. Much of its distinctive and whimsical architecture derives from that period, the river is now home to a yacht club and a sailing club. Aldeburgh is on the North Sea coast and is located around 87 miles north-east of London,20 mi north-east of Ipswich and 23 mi south of Lowestoft, locally it is 4 mi south of the town of Leiston and 2 mi south of the village of Thorpeness. It lies just to the north of the River Alde with the shingle spit of Orford Ness all that stops the river meeting the sea at Aldeburgh - instead it flows another 9 mi to the south-west. The beach is mainly shingle and wide in places with fishing boats able to be drawn up onto the beach above the high tide, but narrows at the neck of Orford Ness. The shingle bank allows access to the Ness from the north, passing a Martello tower, Aldeburgh was flooded during the North Sea flood of 1953 and flood defences around the town were strengthened as a result. The beach was awarded the Blue flag rural beach award in 2005, the town is within the Suffolk Coast and Heaths Area of Outstanding Natural Beauty and has a number of Sites of Special Scientific Interest and nature reserves in the local area. The Alde-Ore Estuary SSSI covers the surrounding the river from Snape to its mouth. This contains a number of salt marsh and mudflat habitats, the Leiston-Aldeburgh SSSI extends from the northern edge of the town to cover a range of habitats including grazing marsh and heathland. It includes Thorpeness Mere and the North Warren RSPB reserve an area of wildlife and habitat conservation, two smaller geological SSSI units are found on the southern edges of the town

2.
Boise, Idaho
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Boise is the capital and most populous city of the U. S. state of Idaho, as well as the county seat of Ada County. Located on the Boise River in southwestern Idaho, the population of Boise at the 2010 Census was 205,671 and its estimated population in 2013 was 214,237. The Boise-Nampa metropolitan area, also known as the Treasure Valley, includes five counties with a population of 664,422. It contains the three largest cities, Boise, Nampa, and Meridian. Boise is the third most populous area in the United States Pacific Northwest region, behind Seattle. Accounts differ about the origin of the name, Bonneville of the U. S. Army as its source. After trekking for weeks through dry and rough terrain, his party reached an overlook with a view of the Boise River Valley. The place where they stood is called Bonneville Point, located on the Oregon Trail east of the city, according to the story, a French-speaking guide, overwhelmed by the sight of the verdant river, yelled Les bois. Les bois. —and the name stuck, the name may instead derive from earlier mountain men, who named the river that flows through it. In the 1820s, French Canadian fur trappers set trap lines in the vicinity, set in a high-desert area, the tree-lined valley of the Boise River became a distinct landmark, an oasis dominated by cottonwood trees. They called this La rivière boisée, which means the wooded river, the area was called Boise long before the establishment of Fort Boise by the federal government. The original Fort Boise was 40 miles west, near Parma and this private sector defense was erected by the Hudsons Bay Company in the 1830s. It was abandoned in the 1850s, but massacres along the Oregon Trail prompted the U. S. Army to re-establish a fort in the area in 1863 during the U. S. Civil War. The new location was selected because it was near the intersection of the Oregon Trail with a road connecting the Boise Basin. During the mid-1860s, Idaho City was the largest city in the Northwest, the original territory was larger than Texas. Mullett, the U. S. Assay Office at 210 Main Street was built in 1871, natives and longtime residents use the pronunciation /ˈbɔɪsiː/, as given on the citys website. The pronunciation is used as a shibboleth, as outsiders tend to pronounce the citys name as /ˈbɔɪziː/. Boise is located in southwestern Idaho, about 41 miles east of the Oregon border, the downtown areas elevation is 2,704 feet above sea level

3.
Time of day
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An hour is a unit of time conventionally reckoned as 1⁄24 of a day and scientifically reckoned as 3, 599–3,601 seconds, depending on conditions. The seasonal, temporal, or unequal hour was established in the ancient Near East as 1⁄12 of the night or daytime, such hours varied by season, latitude, and weather. It was subsequently divided into 60 minutes, each of 60 seconds, the modern English word hour is a development of the Anglo-Norman houre and Middle English ure, first attested in the 13th century. It displaced the Old English tide and stound, the Anglo-Norman term was a borrowing of Old French ure, a variant of ore, which derived from Latin hōra and Greek hṓrā. Like Old English tīd and stund, hṓrā was originally a word for any span of time, including seasons. Its Proto-Indo-European root has been reconstructed as *yeh₁-, making hour distantly cognate with year, the time of day is typically expressed in English in terms of hours. Whole hours on a 12-hour clock are expressed using the contracted phrase oclock, Hours on a 24-hour clock are expressed as hundred or hundred hours. Fifteen and thirty minutes past the hour is expressed as a quarter past or after and half past, respectively, fifteen minutes before the hour may be expressed as a quarter to, of, till, or before the hour. Sumerian and Babylonian hours divided the day and night into 24 equal hours, the ancient Egyptians began dividing the night into wnwt at some time before the compilation of the Dynasty V Pyramid Texts in the 24th century BC. By 2150 BC, diagrams of stars inside Egyptian coffin lids—variously known as diagonal calendars or star clocks—attest that there were exactly 12 of these. The coffin diagrams show that the Egyptians took note of the risings of 36 stars or constellations. Each night, the rising of eleven of these decans were noted, the original decans used by the Egyptians would have fallen noticeably out of their proper places over a span of several centuries. By the time of Amenhotep III, the priests at Karnak were using water clocks to determine the hours and these were filled to the brim at sunset and the hour determined by comparing the water level against one of its twelve gauges, one for each month of the year. During the New Kingdom, another system of decans was used, the later division of the day into 12 hours was accomplished by sundials marked with ten equal divisions. The morning and evening periods when the failed to note time were observed as the first and last hours. The Egyptian hours were closely connected both with the priesthood of the gods and with their divine services, by the New Kingdom, each hour was conceived as a specific region of the sky or underworld through which Ras solar bark travelled. Protective deities were assigned to each and were used as the names of the hours, as the protectors and resurrectors of the sun, the goddesses of the night hours were considered to hold power over all lifespans and thus became part of Egyptian funerary rituals. The Egyptian for astronomer, used as a synonym for priest, was wnwty, the earliest forms of wnwt include one or three stars, with the later solar hours including the determinative hieroglyph for sun

4.
Position of the Sun
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The position of the Sun in the sky is a function of both time and the geographic coordinates of the observer on the surface of the Earth. As the Earth moves around the Sun during the course of the year, the Earths rotation about its axis causes the fixed stars to move in the sky in a way that depends on the observers geographic latitude. The time when a fixed star crosses the observers meridian depends on the geographic longitude. This calculation is useful in astronomy, navigation, surveying, meteorology, climatology, solar energy, start by calculating n, the number of days since Greenwich noon, Terrestrial Time, on 1 January 2000. λ, β and R form a position of the Sun in the ecliptic coordinate system.439 ∘ −0.0000004 ∘ n for use with the above equations. The Sun appears to move northward during the northern spring and its declination reaches a maximum equal to the angle of the Earths axial tilt on the June solstice, then decreases until the December solstice, when its value is the opposite of the axial tilt. A graph of solar declination during a year looks like a wave with an amplitude of 23. 44°. Imagine that the Earth is spherical, in an orbit around the Sun. At one date in the year the Sun would be vertically overhead at the North Pole, eventually the Sun would be over the South Pole, with a declination of -90°. Then it would start to move northward at a constant speed, now suppose that the axial tilt decreases. The absolute maximum and minimum values of the declination would decrease, also, the shapes of the maxima and minima on the graph would become less acute, being curved to resemble the maxima and minima of a sine wave. However, even when the axial tilt equals that of the real Earth, the real Earths orbit is elliptical. The Earth moves more rapidly around the Sun near perihelion, in early January, than near aphelion and this makes processes like the variation of the solar declination happen faster in January than July. On the graph, this makes the minima more acute than the maxima, also, since perihelion and aphelion do not happen on exactly the same dates as the solstices, the maxima and minima are slightly asymmetrical. The rates of change before and after are not quite equal, the graph of apparent solar declination is therefore different in several ways from a sine wave. Calculating it accurately involves some complexity, as shown below, the declination of the Sun, δ☉, is the angle between the rays of the Sun and the plane of the Earths equator. The Earths axial tilt is the angle between the Earths axis and a perpendicular to the Earths orbit. At the solstices, the angle between the rays of the Sun and the plane of the Earths equator reaches its maximum value of 23°26, therefore δ☉ = +23°26 at the northern summer solstice and δ☉ = −23°26 at the southern summer solstice

5.
Sky
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The sky is everything that lies above the surface of the Earth, including the atmosphere and outer space. In the field of astronomy, the sky is called the celestial sphere. This is viewed from Earths surface as a dome where the Sun, stars, planets. The celestial sphere is divided into regions called constellations. Usually, the sky is used informally as the point of view from the Earths surface, however. In some cases, such as in discussing the weather, the sky refers to only the lower, during daylight, the sky appears to be blue because air scatters blue sunlight more than it scatters red. At night, the sky appears to be a dark surface or region scattered with stars. During the day, the Sun can be seen in the sky obscured by clouds. In the night sky the Moon, planets and stars are visible in the sky, some of the natural phenomena seen in the sky are clouds, rainbows, and aurorae. Lightning and precipitation can also be seen in the sky during storms, birds, insects, aircraft, and kites are often considered to fly in the sky. Due to human activities, smog during the day and light pollution during the night are often seen above large cities. Except for light that directly from the sun, most of the light in the day sky is caused by scattering. The scattering due to molecule sized particles is greater in the forward and backward directions than it is in the lateral direction. Scattering is significant for light at all visible wavelengths but is stronger at the end of the visible spectrum, meaning that the scattered light is bluer than its source. The remaining sunlight, having lost some of its wavelength components. Scattering also occurs more strongly in clouds. Individual water droplets exposed to light will create a set of colored rings. If a cloud is thick enough, scattering from water droplets will wash out the set of colored rings

6.
Gnomon
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A gnomon is the part of a sundial that casts a shadow. The term has come to be used for a variety of purposes in mathematics, a painted stick from 2300 BCE found in China is the oldest known gnomon. Anaximander is credited with introducing this Babylonian instrument to the Greeks, oenopides used the phrase drawn gnomon-wise to describe a line drawn perpendicular to another. Later, the term was used for an L-shaped instrument like a steel square used to right angles. This shape may explain its use to describe a shape formed by cutting a square from a larger one. Euclid extended the term to the figure formed by removing a similar parallelogram from a corner of a larger parallelogram. Indeed, the gnomon is the increment between two successive figurate numbers, including square and triangular numbers, hero of Alexandria defined a gnomon as that which, when added to an entity, makes a new entity similar to the starting entity. In this sense Theon of Smyrna used it to describe a number added to a polygonal number produces the next one of the same type. The most common use in this sense is an odd integer especially when seen as a number between square numbers. Perforated gnomons projecting an image of the sun were described in the Chinese Zhoubi Suanjing writings. The location of the circle can be measured to tell the time of day. In Arab and European cultures its invention was later attributed to Egyptian astronomer. With markings on the floor it tells the time of each midday as well as the date of the summer solstice. Italian mathematician, engineer, astronomer and geographer Leonardo Ximenes reconstructed the gnomon according to his new measurements in 1756, in the northern hemisphere, the shadow-casting edge of a sundial gnomon is normally oriented so that it points north and is parallel to the rotation axis of the Earth. That is, it is inclined to the horizontal at an angle equals the latitude of the sundials location. At present, such a gnomon should thus point almost precisely at Polaris, on some sundials, the gnomon is vertical. These were usually used in times for observing the altitude of the Sun. The style is the part of the gnomon that casts the shadow and this can change as the sun moves

7.
Shadow
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A shadow is a dark area where light from a light source is blocked by an opaque object. It occupies all of the volume behind an object with light in front of it. The cross section of a shadow is a silhouette, or a reverse projection of the object blocking the light. A point source of light casts only a shadow, called an umbra. For a non-point or extended source of light, the shadow is divided into the umbra, penumbra and antumbra, the wider the light source, the more blurred the shadow becomes. If two penumbras overlap, the appear to attract and merge. This is known as the Shadow Blister Effect, the outlines of the shadow zones can be found by tracing the rays of light emitted by the outermost regions of the extended light source. The umbra region does not receive any direct light from any part of the light source, a viewer located in the umbra region cannot directly see any part of the light source. By contrast, the penumbra is illuminated by some parts of the light source, a viewer located in the penumbra region will see the light source, but it is partially blocked by the object casting the shadow. If there is more than one source, there will be several shadows, with the overlapping parts darker. The more diffuse the lighting is, the softer and more indistinct the shadow outlines become, the lighting of an overcast sky produces few visible shadows. The absence of diffusing atmospheric effects in the vacuum of outer space produces shadows that are stark, for a person or object touching the surface where the shadow is projected the shadows converge at the point of contact. A shadow shows, apart from distortion, the image as the silhouette when looking at the object from the sun-side. The names umbra, penumbra and antumbra are often used for the shadows cast by objects, though they are sometimes used to describe levels of darkness. An astronomical object casts human-visible shadows when its apparent magnitude is equal or lower than -4, currently the only astronomical objects able to produce visible shadows on Earth are the sun, the moon and, in the right conditions, Venus or Jupiter. A shadow cast by the Earth on the Moon is a lunar eclipse, conversely, a shadow cast by the Moon on the Earth is a solar eclipse. The sun casts shadows which change dramatically through the day, the length of a shadow cast on the ground is proportional to the cotangent of the suns elevation angle—its angle θ relative to the horizon. Near sunrise and sunset, when θ = 0° and cot = ∞, if the sun passes directly overhead, then θ = 90°, cot =0, and shadows are cast directly underneath objects

8.
Sun
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The Sun is the star at the center of the Solar System. It is a perfect sphere of hot plasma, with internal convective motion that generates a magnetic field via a dynamo process. It is by far the most important source of energy for life on Earth. Its diameter is about 109 times that of Earth, and its mass is about 330,000 times that of Earth, accounting for about 99. 86% of the total mass of the Solar System. About three quarters of the Suns mass consists of hydrogen, the rest is mostly helium, with smaller quantities of heavier elements, including oxygen, carbon, neon. The Sun is a G-type main-sequence star based on its spectral class and it formed approximately 4.6 billion years ago from the gravitational collapse of matter within a region of a large molecular cloud. Most of this matter gathered in the center, whereas the rest flattened into a disk that became the Solar System. The central mass became so hot and dense that it eventually initiated nuclear fusion in its core and it is thought that almost all stars form by this process. The Sun is roughly middle-aged, it has not changed dramatically for more than four billion years and it is calculated that the Sun will become sufficiently large enough to engulf the current orbits of Mercury, Venus, and probably Earth. The enormous effect of the Sun on Earth has been recognized since prehistoric times, the synodic rotation of Earth and its orbit around the Sun are the basis of the solar calendar, which is the predominant calendar in use today. The English proper name Sun developed from Old English sunne and may be related to south, all Germanic terms for the Sun stem from Proto-Germanic *sunnōn. The English weekday name Sunday stems from Old English and is ultimately a result of a Germanic interpretation of Latin dies solis, the Latin name for the Sun, Sol, is not common in general English language use, the adjectival form is the related word solar. The term sol is used by planetary astronomers to refer to the duration of a solar day on another planet. A mean Earth solar day is approximately 24 hours, whereas a mean Martian sol is 24 hours,39 minutes, and 35.244 seconds. From at least the 4th Dynasty of Ancient Egypt, the Sun was worshipped as the god Ra, portrayed as a falcon-headed divinity surmounted by the solar disk, and surrounded by a serpent. In the New Empire period, the Sun became identified with the dung beetle, in the form of the Sun disc Aten, the Sun had a brief resurgence during the Amarna Period when it again became the preeminent, if not only, divinity for the Pharaoh Akhenaton. The Sun is viewed as a goddess in Germanic paganism, Sól/Sunna, in ancient Roman culture, Sunday was the day of the Sun god. It was adopted as the Sabbath day by Christians who did not have a Jewish background, the symbol of light was a pagan device adopted by Christians, and perhaps the most important one that did not come from Jewish traditions

9.
Earth's rotation
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Earths rotation is the rotation of the planet Earth around its own axis. The Earth rotates from the west towards east, as viewed from North Star or polestar Polaris, the Earth turns counter-clockwise. The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earths axis of rotation meets its surface and this point is distinct from the Earths North Magnetic Pole. The South Pole is the point where the Earths axis of rotation intersects its surface. The Earth rotates once in about 24 hours with respect to the sun, Earths rotation is slowing slightly with time, thus, a day was shorter in the past. This is due to the effects the Moon has on Earths rotation. Atomic clocks show that a modern-day is longer by about 1.7 milliseconds than a century ago, analysis of historical astronomical records shows a slowing trend of 2.3 milliseconds per century since the 8th century BCE. Among the ancient Greeks, several of the Pythagorean school believed in the rotation of the rather than the apparent diurnal rotation of the heavens. Perhaps the first was Philolaus, though his system was complicated, in the third century BCE, Aristarchus of Samos suggested the suns central place. However, Aristotle in the fourth century criticized the ideas of Philolaus as being based on rather than observation. He established the idea of a sphere of fixed stars that rotated about the earth and this was accepted by most of those who came after, in particular Claudius Ptolemy, who thought the earth would be devastated by gales if it rotated. In the 10th century, some Muslim astronomers accepted that the Earth rotates around its axis, treatises were written to discuss its possibility, either as refutations or expressing doubts about Ptolemys arguments against it. At the Maragha and Samarkand observatories, the Earths rotation was discussed by Tusi and Qushji, in medieval Europe, Thomas Aquinas accepted Aristotles view and so, reluctantly, did John Buridan and Nicole Oresme in the fourteenth century. Not until Nicolaus Copernicus in 1543 adopted a heliocentric world system did the earths rotation begin to be established, Copernicus pointed out that if the movement of the earth is violent, then the movement of the stars must be very much more so. He acknowledged the contribution of the Pythagoreans and pointed to examples of relative motion, for Copernicus this was the first step in establishing the simpler pattern of planets circling a central sun. Tycho Brahe, who produced accurate observations on which Kepler based his laws, in 1600, William Gilbert strongly supported the earths rotation in his treatise on the earths magnetism and thereby influenced many of his contemporaries. Those like Gilbert who did not openly support or reject the motion of the earth about the sun are often called semi-Copernicans, however, the contributions of Kepler, Galileo and Newton gathered support for the theory of the rotation of the Earth. The earths rotation implies that the bulges and the poles are flattened

10.
Latitude
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In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earths surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles, lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the location of features on the surface of the Earth. Without qualification the term latitude should be taken to be the latitude as defined in the following sections. Also defined are six auxiliary latitudes which are used in special applications, there is a separate article on the History of latitude measurements. Two levels of abstraction are employed in the definition of latitude and longitude, in the first step the physical surface is modelled by the geoid, a surface which approximates the mean sea level over the oceans and its continuation under the land masses. The second step is to approximate the geoid by a mathematically simpler reference surface, the simplest choice for the reference surface is a sphere, but the geoid is more accurately modelled by an ellipsoid. The definitions of latitude and longitude on such surfaces are detailed in the following sections. Lines of constant latitude and longitude together constitute a graticule on the reference surface, latitude and longitude together with some specification of height constitute a geographic coordinate system as defined in the specification of the ISO19111 standard. This is of importance in accurate applications, such as a Global Positioning System, but in common usage, where high accuracy is not required. In English texts the latitude angle, defined below, is denoted by the Greek lower-case letter phi. It is measured in degrees, minutes and seconds or decimal degrees, the precise measurement of latitude requires an understanding of the gravitational field of the Earth, either to set up theodolites or to determine GPS satellite orbits. The study of the figure of the Earth together with its field is the science of geodesy. These topics are not discussed in this article and this article relates to coordinate systems for the Earth, it may be extended to cover the Moon, planets and other celestial objects by a simple change of nomenclature. The primary reference points are the poles where the axis of rotation of the Earth intersects the reference surface, the plane through the centre of the Earth and perpendicular to the rotation axis intersects the surface at a great circle called the Equator. Planes parallel to the plane intersect the surface in circles of constant latitude. The Equator has a latitude of 0°, the North Pole has a latitude of 90° North, the latitude of an arbitrary point is the angle between the equatorial plane and the radius to that point. The latitude, as defined in this way for the sphere, is termed the spherical latitude

11.
Horizontal coordinate system
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The horizontal coordinate system is a celestial coordinate system that uses the observers local horizon as the fundamental plane. It is expressed in terms of angle and azimuth. This coordinate system divides the sky into the upper hemisphere where objects are visible, the great circle separating the hemispheres is called the celestial horizon. The celestial horizon is defined as the circle on the celestial sphere whose plane is normal to the local gravity vector. In practice, the horizon can be defined as the tangent to a still liquid surface such as a pool of mercury. The pole of the upper hemisphere is called the zenith, the pole of the lower hemisphere is called the nadir. There are two independent horizontal angular coordinates, Altitude, sometimes referred to as elevation, is the angle between the object and the local horizon. For visible objects it is an angle between 0 degrees to 90 degrees, alternatively, zenith distance may be used instead of altitude. More details on the computation of azimuth and zenith angle can be found at Solar azimuth angle, the horizontal coordinate system is fixed to the Earth, not the stars. Therefore, the altitude and azimuth of an object in the sky changes with time, horizontal coordinates are very useful for determining the rise and set times of an object in the sky. When an objects altitude is 0°, it is on the horizon, if at that moment its altitude is increasing, it is rising, but if its altitude is decreasing, it is setting. However, all objects on the sphere are subject to diurnal motion. One can determine whether altitude is increasing or decreasing by instead considering the azimuth of the celestial object, if the azimuth is between 180° and 360°, it is setting. There are the special cases, As seen from the north pole all directions are south. Viewed from either pole, a star has constant altitude, the Sun, Moon, and planets can rise or set over the span of a year when viewed from the poles because their declinations are constantly changing. As seen from the equator, objects on the poles stay at fixed points on the horizon. Note that the above considerations are strictly speaking true for the horizon only. That is, the horizon as it would appear for an observer at sea level on a perfectly smooth Earth without an atmosphere

12.
Azimuth
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An azimuth is an angular measurement in a spherical coordinate system. An example is the position of a star in the sky, the star is the point of interest, the reference plane is the horizon or the surface of the sea, and the reference vector points north. The azimuth is the angle between the vector and the perpendicular projection of the star down onto the horizon. Azimuth is usually measured in degrees, the concept is used in navigation, astronomy, engineering, mapping, mining and artillery. In land navigation, azimuth is usually denoted alpha, α, azimuth has also been more generally defined as a horizontal angle measured clockwise from any fixed reference plane or easily established base direction line. Today, the plane for an azimuth is typically true north, measured as a 0° azimuth. Moving clockwise on a 360 degree circle, east has azimuth 90°, south 180°, there are exceptions, some navigation systems use south as the reference vector. Any direction can be the vector, as long as it is clearly defined. Quite commonly, azimuths or compass bearings are stated in a system in which either north or south can be the zero, the reference direction, stated first, is always north or south, and the turning direction, stated last, is east or west. The directions are chosen so that the angle, stated between them, is positive, between zero and 90 degrees. If the bearing happens to be exactly in the direction of one of the cardinal points and this is the reason why the X and Y axis in the above formula are swapped. If the azimuth becomes negative, one can always add 360°, the formula in radians would be slightly easier, α = atan2 ⁡ Caveat, Most computer libraries reverse the order of the atan2 parameters. We are standing at latitude φ1, longitude zero, we want to find the azimuth from our viewpoint to Point 2 at latitude φ2, longitude L. The difference is usually small, if Point 2 is not more than 100 km away. Various websites will calculate geodetic azimuth, e. g. GeoScience Australia site, formulas for calculating geodetic azimuth are linked in the distance article. Normal-section azimuth is simpler to calculate, Bomford says Cunninghams formula is exact for any distance, replace φ2 with declination and longitude difference with hour angle, and change the sign. There is a variety of azimuthal map projections. They all have the property that directions from a point are preserved

13.
Cylindrical lens
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A cylindrical lens is a lens which focuses light into a line instead of a point, as a spherical lens would. The lens compresses the image in the perpendicular to this line. In a light microscope, a cylindrical lens is placed in front of the illumination objective to create the light sheet used for imaging. Lenticular lens Jacobs, Donald H. Fundamentals of Optical Engineering

14.
Plane (mathematics)
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In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the analogue of a point, a line. When working exclusively in two-dimensional Euclidean space, the article is used, so. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a space, or in other words. Euclid set forth the first great landmark of mathematical thought, a treatment of geometry. He selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its sense is not directly given a definition anywhere in the Elements. In his work Euclid never makes use of numbers to measure length, angle, in this way the Euclidean plane is not quite the same as the Cartesian plane. This section is concerned with planes embedded in three dimensions, specifically, in R3. In a Euclidean space of any number of dimensions, a plane is determined by any of the following. A line and a point not on that line, a line is either parallel to a plane, intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the plane must be parallel to each other. Two distinct planes perpendicular to the line must be parallel to each other. Specifically, let r0 be the vector of some point P0 =. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the plane can be described as the set of all points r such that n ⋅ =0. Expanded this becomes a + b + c =0, which is the form of the equation of a plane. This is just a linear equation a x + b y + c z + d =0 and this familiar equation for a plane is called the general form of the equation of the plane

15.
Sphere
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A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twice the radius. While outside mathematics the terms sphere and ball are used interchangeably. The ball and the share the same radius, diameter. The surface area of a sphere is, A =4 π r 2, at any given radius r, the incremental volume equals the product of the surface area at radius r and the thickness of a shell, δ V ≈ A ⋅ δ r. The total volume is the summation of all volumes, V ≈ ∑ A ⋅ δ r. In the limit as δr approaches zero this equation becomes, V = ∫0 r A d r ′, substitute V,43 π r 3 = ∫0 r A d r ′. Differentiating both sides of equation with respect to r yields A as a function of r,4 π r 2 = A. Which is generally abbreviated as, A =4 π r 2, alternatively, the area element on the sphere is given in spherical coordinates by dA = r2 sin θ dθ dφ. In Cartesian coordinates, the element is d S = r r 2 − ∑ i ≠ k x i 2 ∏ i ≠ k d x i, ∀ k. For more generality, see area element, the total area can thus be obtained by integration, A = ∫02 π ∫0 π r 2 sin ⁡ θ d θ d φ =4 π r 2. In three dimensions, the volume inside a sphere is derived to be V =43 π r 3 where r is the radius of the sphere, archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. In modern mathematics, this formula can be derived using integral calculus, at any given x, the incremental volume equals the product of the cross-sectional area of the disk at x and its thickness, δ V ≈ π y 2 ⋅ δ x. The total volume is the summation of all volumes, V ≈ ∑ π y 2 ⋅ δ x. In the limit as δx approaches zero this equation becomes, V = ∫ − r r π y 2 d x. At any given x, a right-angled triangle connects x, y and r to the origin, hence, applying the Pythagorean theorem yields, thus, substituting y with a function of x gives, V = ∫ − r r π d x. Which can now be evaluated as follows, V = π − r r = π − π =43 π r 3

16.
Cylinder (geometry)
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In its simplest form, a cylinder is the surface formed by the points at a fixed distance from a given straight line called the axis of the cylinder. It is one of the most basic curvilinear geometric shapes, commonly the word cylinder is understood to refer to a finite section of a right circular cylinder having a finite height with circular ends perpendicular to the axis as shown in the figure. If the ends are open, it is called an open cylinder, if the ends are closed by flat surfaces it is called a solid cylinder. The formulae for the area and the volume of such a cylinder have been known since deep antiquity. The area of the side is known as the lateral area. An open cylinder does not include either top or bottom elements, the surface area of a closed cylinder is made up the sum of all three components, top, bottom and side. Its surface area is A = 2πr2 + 2πrh = 2πr = πd=L+2B, for a given volume, the closed cylinder with the smallest surface area has h = 2r. Equivalently, for a surface area, the closed cylinder with the largest volume has h = 2r. Cylindric sections are the intersections of cylinders with planes, for a right circular cylinder, there are four possibilities. A plane tangent to the cylinder meets the cylinder in a straight line segment. Moved while parallel to itself, the plane either does not intersect the cylinder or intersects it in two line segments. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, a cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively. Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plücker conoid. The volume of a cylinder with height h is V = ∫0 h A d x = ∫0 h π a b d x = π a b ∫0 h d x = π a b h. Even more general than the cylinder is the generalized cylinder. The cylinder is a degenerate quadric because at least one of the coordinates does not appear in the equation, an oblique cylinder has the top and bottom surfaces displaced from one another. There are other unusual types of cylinders. Let the height be h, internal radius r, and external radius R, the volume is given by V = π h

17.
Cone (geometry)
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A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex. A cone is formed by a set of segments, half-lines, or lines connecting a common point. If the enclosed points are included in the base, the cone is a solid object, otherwise it is a two-dimensional object in three-dimensional space. In the case of an object, the boundary formed by these lines or partial lines is called the lateral surface, if the lateral surface is unbounded. In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, in the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a cone on one side of the apex is called a nappe. The axis of a cone is the line, passing through the apex. If the base is right circular the intersection of a plane with this surface is a conic section, in general, however, the base may be any shape and the apex may lie anywhere. Contrasted with right cones are oblique cones, in which the axis passes through the centre of the base non-perpendicularly, a cone with a polygonal base is called a pyramid. Depending on the context, cone may also mean specifically a convex cone or a projective cone, cones can also be generalized to higher dimensions. The perimeter of the base of a cone is called the directrix, the base radius of a circular cone is the radius of its base, often this is simply called the radius of the cone. The aperture of a circular cone is the maximum angle between two generatrix lines, if the generatrix makes an angle θ to the axis, the aperture is 2θ. A cone with a region including its apex cut off by a plane is called a cone, if the truncation plane is parallel to the cones base. An elliptical cone is a cone with an elliptical base, a generalized cone is the surface created by the set of lines passing through a vertex and every point on a boundary. The slant height of a circular cone is the distance from any point on the circle to the apex of the cone. It is given by r 2 + h 2, where r is the radius of the cirf the cone and this application is primarily useful in determining the slant height of a cone when given other information regarding the radius or height. The volume V of any conic solid is one third of the product of the area of the base A B and the height h V =13 A B h. In modern mathematics, this formula can easily be computed using calculus – it is, up to scaling, the integral ∫ x 2 d x =13 x 3

18.
Analemmatic sundial
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Analemmatic sundials are a type of horizontal sundial that has a vertical gnomon and hour markers positioned in an elliptical pattern. The gnomon is not fixed and must change position daily to indicate time of day. Hence there are no lines on the dial and the time of day is read only on the ellipse. As with most sundials, analemmatic sundials mark solar time rather than clock time, an analemmatic sundial is completely defined by The size of its ellipse. Analemmatic sundials are designed with a human as the gnomon. Human gnomon analemmatic sundials are not practical at lower latitudes where a shadow is quite short during the summer months. A66 inch tall person casts a 4-inch shadow at 27 deg latitude on the summer solstice. The use of the adjective analemmatic to describe this class of sundial can be misleading, mayall refers to the analemmatic sundial as the so-called Analemmatic Dial, implying a lack of connection to the analemma. The dial of Brou in front of the church of Brou in Bourg-en-Bresse, rohr states The gnomon is displaced on the short axis of the ellipse and not on the meridian, whose presence here in the shape of an 8 is a mistake. An analemmatic sundial uses a vertical gnomon and its lines are the vertical projection of the hour lines of a circular equatorial sundial onto a flat plane. Therefore, the sundial is an ellipse, where the short axis is aligned North-South. The declination measures how far the sun is above the equator, at the equinoxes. Analemma Analemmatic sundial generator Analemmatic and Horizontal Sundials of the Bronze Age History and principle of analemmatic sundials

19.
Solar time
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Solar time is a calculation of the passage of time based on the position of the Sun in the sky. The fundamental unit of time is the day. Two types of time are apparent solar time and mean solar time. Fix a tall pole vertically in the ground, at some instant on any day the shadow will point exactly north or south. That instant is local apparent noon,12,00 local apparent time, about 24 hours later the shadow will again point north/south, the Sun seeming to have covered a 360-degree arc around the Earths axis. When the Sun has covered exactly 15 degrees, local apparent time is 13,00 exactly, after 15 more degrees it will be 14,00 exactly. As explained in the equation of time article, this is due to the eccentricity of the Earths orbit, the effect of this is that a clock running at a constant rate – e. g. This is mean solar time, which is not perfectly constant from one century to the next but is close enough for most purposes. Currently a mean solar day is about 86,400.002 SI seconds, the two kinds of solar time are among the three kinds of time reckoning that were employed by astronomers until the 1950s. By the 1950s it had become clear that the Earths rotation rate was not constant, so astronomers developed ephemeris time, the apparent sun is the true sun as seen by an observer on Earth. Apparent solar time or true solar time is based on the apparent motion of the actual Sun and it is based on the apparent solar day, the interval between two successive returns of the Sun to the local meridian. Solar time can be measured by a sundial. The equivalent on other planets is termed local true solar time, the length of a solar day varies through the year, and the accumulated effect produces seasonal deviations of up to 16 minutes from the mean. The effect has two main causes, first, Earths orbit is an ellipse, not a circle, so the Earth moves faster when it is nearest the Sun and slower when it is farthest from the Sun. Second, due to Earths axial tilt, the Suns annual motion is along a circle that is tilted to Earths celestial equator. In June and December when the sun is farthest from the equator a given shift along the ecliptic corresponds to a large shift at the equator. So apparent solar days are shorter in March and September than in June or December and these lengths will change slightly in a few years and significantly in thousands of years. Mean solar time is the angle of the mean Sun plus 12 hours

20.
Equation of time
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The equation of time describes the discrepancy between two kinds of solar time. The word equation is used in the sense of reconcile a difference. The two times that differ are the apparent solar time, which tracks the motion of the sun, and mean solar time. Apparent solar time can be obtained by measurement of the current position of the Sun, Mean solar time, for the same place, would be the time indicated by a steady clock set so that over the year its differences from apparent solar time would resolve to zero. The equation of time is the east or west component of the analemma, the equation of time values for each day of the year, compiled by astronomical observatories, were widely listed in almanacs and ephemerides. During a year the equation of time varies as shown on the graph, apparent time, and the sundial, can be ahead by as much as 16 min 33 s, or behind by as much as 14 min 6 s. The equation of time has zeros near 15 April,13 June,1 September and 25 December, ignoring very slow changes in the Earths orbit and rotation, these events are repeated at the same times every tropical year. However, due to the number of days in a year. The equation of time is constant only for a planet with zero axial tilt, on Mars the difference between sundial time and clock time can be as much as 50 minutes, due to the considerably greater eccentricity of its orbit. The planet Uranus, which has a large axial tilt, has an equation of time that makes its days start. There is no accepted definition of the sign of the equation of time. Some publications show it as positive when a sundial is ahead of a clock, as shown in the graph above, others when the clock is ahead of the sundial. In the English-speaking world, the usage is the more common. Anyone who makes use of a table or graph should first check its sign usage. Often, there is a note or caption which explains it, otherwise, the sign can be determined by knowing that, during the first three months of each year, the clock is ahead of the sundial. The mnemonic NYSS, for new year, sundial slow, can be useful, some published tables avoid the ambiguity by not using signs, but by showing phrases such as sundial fast or sundial slow instead. In this article, and others in English Wikipedia, a value of the equation of time implies that a sundial is ahead of a clock. The phrase equation of time is derived from the medieval Latin, the word aequātiō was widely used in early astronomy to tabulate the difference between an observed value and the expected value

21.
Greenwich
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Greenwich is an early-established district of todays London, England, centred 5.5 miles east south-east of Charing Cross. The town lends its name to the Royal Borough of Greenwich, Greenwich is generally described as being part of South-east London and sometimes as being part of East London. Greenwich is notable for its history and for giving its name to the Greenwich Meridian. The town became the site of a palace, the Palace of Placentia from the 15th century. The palace fell into disrepair during the English Civil War and was rebuilt as the Royal Naval Hospital for Sailors by Sir Christopher Wren and his assistant Nicholas Hawksmoor. These buildings became the Royal Naval College in 1873, and they remained an establishment for military education until 1998 when they passed into the hands of the Greenwich Foundation. The historic rooms within these buildings remain open to the public, other buildings are used by University of Greenwich and Trinity Laban Conservatoire of Music and Dance. The town became a resort in the 18th century and many grand houses were built there, such as Vanbrugh Castle established on Maze Hill. From the Georgian period estates of houses were constructed above the town centre, Greenwich formed part of Kent until 1889 when the County of London was created. The place-name Greenwich is first attested in a Saxon charter of 918 and it is recorded as Grenewic in 964, and as Grenawic in the Anglo-Saxon Chronicle for 1013. It is Grenviz in the Domesday Book of 1086, and Grenewych in the Taxatio Ecclesiastica of 1291, the name means green wic or settlement. An article in The Times of 13 October 1967 stated, East Greenwich, gateway to the Blackwall Tunnel, remains solidly working class, the manpower for one eighth of Londons heavy industry. West Greenwich is a hybrid, the spirit of Nelson, the Cutty Sark, the Maritime Museum, an industrial waterfront and a number of elegant houses, ripe for development. Royal charters granted to English colonists in North America, often used the name of the manor of East Greenwich for describing the tenure as that of free socage, New England charters provided that the grantees should hold their lands as of his Majesty’s manor of East Greenwich. Grants named the castle of Windsor, places in North America that have taken the name East Greenwich include a township in Gloucester County, New Jersey, a hamlet in Washington County, New York, and a town in Kent County, Rhode Island. Tumuli to the south-west of Flamsteed House, in Greenwich Park, are thought to be early Bronze Age barrows re-used by the Saxons in the 6th century as burial grounds, to the east between the Vanbrugh and Maze Hill Gates is the site of a Roman villa or temple. A small area of red paving tesserae protected by railings marks the spot and it was excavated in 1902 and 300 coins were found dating from the emperors Claudius and Honorius to the 5th century. This was excavated by the Channel 4 television programme Time Team in 1999, broadcast in 2000, the Roman road from London to Dover, Watling Street crossed the high ground to the south of Greenwich, through Blackheath

22.
Daylight saving time
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Daylight saving time is the practice of advancing clocks during summer months by one hour so that evening daylight lasts an hour longer, while sacrificing normal sunrise times. Typically, regions that use Daylight Savings Time adjust clocks forward one hour close to the start of spring, American inventor and politician Benjamin Franklin proposed a form of daylight time in 1784. New Zealander George Hudson proposed the idea of saving in 1895. The German Empire and Austria-Hungary organized the first nationwide implementation, starting on April 30,1916, many countries have used it at various times since then, particularly since the energy crisis of the 1970s. The practice has both advocates and critics, DST clock shifts sometimes complicate timekeeping and can disrupt travel, billing, record keeping, medical devices, heavy equipment, and sleep patterns. Computer software often adjusts clocks automatically, but policy changes by various jurisdictions of DST dates, industrialized societies generally follow a clock-based schedule for daily activities that do not change throughout the course of the year. The time of day that individuals begin and end work or school, North and south of the tropics daylight lasts longer in summer and shorter in winter, with the effect becoming greater as one moves away from the tropics. However, they will have one hour of daylight at the start of each day. Supporters have also argued that DST decreases energy consumption by reducing the need for lighting and heating, DST is also of little use for locations near the equator, because these regions see only a small variation in daylight in the course of the year. After ancient times, equal-length civil hours eventually supplanted unequal, so civil time no longer varies by season, unequal hours are still used in a few traditional settings, such as some monasteries of Mount Athos and all Jewish ceremonies. This 1784 satire proposed taxing window shutters, rationing candles, and waking the public by ringing church bells, despite common misconception, Franklin did not actually propose DST, 18th-century Europe did not even keep precise schedules. However, this changed as rail transport and communication networks came to require a standardization of time unknown in Franklins day. Modern DST was first proposed by the New Zealand entomologist George Hudson, whose shift work job gave him time to collect insects. An avid golfer, he also disliked cutting short his round at dusk and his solution was to advance the clock during the summer months, a proposal he published two years later. The proposal was taken up by the Liberal Member of Parliament Robert Pearce, a select committee was set up to examine the issue, but Pearces bill did not become law, and several other bills failed in the following years. Willett lobbied for the proposal in the UK until his death in 1915, william Sword Frost, mayor of Orillia, Ontario, introduced daylight saving time in the municipality during his tenure from 1911 to 1912. Starting on April 30,1916, the German Empire and its World War I ally Austria-Hungary were the first to use DST as a way to conserve coal during wartime, Britain, most of its allies, and many European neutrals soon followed suit. Russia and a few other countries waited until the year

23.
Aberdeen, Scotland
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Nicknames include the Granite City, the Grey City and the Silver City with the Golden Sands. During the mid-18th to mid-20th centuries, Aberdeens buildings incorporated locally quarried grey granite, since the discovery of North Sea oil in the 1970s, other nicknames have been the Oil Capital of the World or the Energy Capital of the World. The area around Aberdeen has been settled since at least 8,000 years ago, the city has a long, sandy coastline and a marine climate, the latter resulting in chilly summers and mild winters. Aberdeen received Royal Burgh status from David I of Scotland, transforming the city economically, the traditional industries of fishing, paper-making, shipbuilding, and textiles have been overtaken by the oil industry and Aberdeens seaport. Aberdeen Heliport is one of the busiest commercial heliports in the world, in 2015, Mercer named Aberdeen the 57th most liveable city in the world, as well as the fourth most liveable city in Britain. In 2012, HSBC named Aberdeen as a business hub and one of eight super cities spearheading the UKs economy. The Aberdeen area has seen human settlement for at least 8,000 years. The city began as two separate burghs, Old Aberdeen at the mouth of the river Don, and New Aberdeen, a fishing and trading settlement, the earliest charter was granted by William the Lion in 1179 and confirmed the corporate rights granted by David I. In 1319, the Great Charter of Robert the Bruce transformed Aberdeen into a property-owning, granted with it was the nearby Forest of Stocket, whose income formed the basis for the citys Common Good Fund which still benefits Aberdonians. The city was burned by Edward III of England in 1336, but was rebuilt and extended, the city was strongly fortified to prevent attacks by neighbouring lords, but the gates were removed by 1770. During the Wars of the Three Kingdoms of 1644–1647 the city was plundered by both sides, in 1644, it was taken and ransacked by Royalist troops after the Battle of Aberdeen and two years later it was stormed by a Royalist force under the command of Marquis of Huntly. In 1647 an outbreak of plague killed a quarter of the population. In the 18th century, a new Town Hall was built and the first social services appeared with the Infirmary at Woolmanhill in 1742 and the Lunatic Asylum in 1779. The council began major road improvements at the end of the 18th century with the main thoroughfares of George Street, King Street, gas street lighting arrived in 1824 and an enhanced water supply appeared in 1830 when water was pumped from the Dee to a reservoir in Union Place. An underground sewer system replaced open sewers in 1865, the city was incorporated in 1891. Although Old Aberdeen has a history and still holds its ancient charter. It is an part of the city, as is Woodside. Old Aberdeen is the location of Aberdon, the first settlement of Aberdeen

24.
Sitka, Alaska
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As of the 2010 census, the population was 8,881. In terms of area, it is the largest city-borough in the U. S. with a land area of 2,870.3 square miles. Urban Sitka, the part that is thought of as the city of Sitka, is on the west side of Baranof Island. The current name Sitka means People on the Outside of Baranof Island, Sitkas location was originally settled by the Tlingit people over 10,000 years ago. The Russians settled Old Sitka in 1799, calling it Redoubt Saint Michael, the governor of Russian America, Alexander Baranov, arrived under the auspices of the Russian-American Company, a colonial trading company chartered by Tsar Paul I. In June 1802, Tlingit warriors destroyed the settlement, killing many of the Russians. Baranov was forced to levy 10,000 rubles in ransom for the return of the surviving settlers. Baranov returned to Sitka in August 1804, with a large force, the ship bombarded the Tlingit fort on the 20th, but was not able to cause significant damage. The Russians then launched an attack on the fort and were repelled, however, after two days of bombardment, the Tlingit hung out a white flag on the 22nd, and then deserted the fort on the 26th. Following their victory at the Battle of Sitka, the Russians established New Archangel as a permanent settlement named after Arkhangelsk, the Tlingit re-established a fort on the Chatham Strait side of Peril Strait to enforce a trade embargo with the Russian establishment. In 1808, with Baranov still governor, Sitka was designated the capital of Russian America, bishop Innocent lived in Sitka after 1840. The Cathedral of Saint Michael was built in Sitka in 1848 and became the seat of the Russian Orthodox bishop of Kamchatka, the Kurile and Aleutian Islands, and Alaska. Swedes, Finns and other Lutherans worked for the Russian-American Company, Sitka was the site of the transfer ceremony for the Alaska purchase on October 18,1867. Russia was going through economic and political turmoil after it lost the Crimean War to Britain, France, russia offered to sell it to the United States. Secretary of State William Seward had wanted to purchase Alaska for quite time as he saw it as an integral part of Manifest Destiny. While the agreement to purchase Alaska was made in April 1867, the cost to purchase Alaska was $7.2 million. Sitka served as the capital of the Alaska Territory until 1906, the Alaska Native Brotherhood was founded in Sitka in 1912 to address racism against Alaska Native people in Alaska. By 1914 the organization had constructed the Alaska Native Brotherhood Hall on Katlian Street, in 1937, the United States Navy established the first seaplane base in Alaska on Japonski Island

25.
Celestial sphere
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In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with Earth. All objects in the sky can be thought of as projected upon the inside surface of the celestial sphere. The celestial sphere is a tool for spherical astronomy, allowing observers to plot positions of objects in the sky when their distances are unknown or unimportant. Because astronomical objects are at such distances, casual observation of the sky offers no information on the actual distances. All objects seem equally far away, as if fixed to the inside of a sphere of large but unknown radius, which rotates from east to west overhead while underfoot, the celestial sphere can be considered to be infinite in radius. This means any point within it, including that occupied by the observer, all parallel planes will seem to intersect the sphere in a coincident great circle. On an infinite-radius celestial sphere, all observers see the things in the same direction. For some objects, this is over-simplified, objects which are relatively near to the observer will seem to change position against the distant celestial sphere if the observer moves far enough, say, from one side of the Earth to the other. This effect, known as parallax, can be represented as an offset from a mean position. The celestial sphere can be considered to be centered at the Earths center, the Suns center, or any convenient location. Individual observers can work out their own small offsets from the mean positions, in many cases in astronomy, the offsets are insignificant. The celestial sphere can thus be thought of as a kind of astronomical shorthand, for many rough uses, this position, as seen from the Earths center, is adequate. This greatly abbreviates the amount of detail necessary in such almanacs and these concepts are important for understanding celestial coordinate systems – frameworks for measuring the positions of objects in the sky. Certain reference lines and planes on Earth, when projected onto the celestial sphere and these include the Earths equator, axis, and the Earths orbit. At their intersections with the sphere, these form the celestial equator, the north and south celestial poles. As the celestial sphere is considered infinite in radius, all observers see the celestial equator, celestial poles, from these bases, directions toward objects in the sky can be quantified by constructing celestial coordinate systems. Similar to terrestrial longitude and latitude, the coordinate system specifies positions relative to the celestial equator and celestial poles. The ecliptic coordinate system specifies positions relative to the Earths orbit, besides the equatorial and ecliptic systems, some other celestial coordinate systems, such as the galactic coordinate system, are more appropriate for particular purposes

26.
Celestial pole
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The north and south celestial poles are the two imaginary points in the sky where the Earths axis of rotation, indefinitely extended, intersects the celestial sphere. The north and south celestial poles appear permanently directly overhead to an observer at the Earths North Pole and South Pole respectively. As the Earth spins on its axis, the two celestial poles remain fixed in the sky, and all other points appear to rotate around them, completing one circuit per day. The celestial poles are also the poles of the equatorial coordinate system, meaning they have declinations of +90 degrees. The celestial poles do not remain permanently fixed against the background of the stars, because of a phenomenon known as the precession of the equinoxes, the poles trace out circles on the celestial sphere, with a period of about 25,700 years. The Earths axis is subject to other complex motions which cause the celestial poles to shift slightly over cycles of varying lengths, see nutation, polar motion. Finally, over long periods the positions of the stars themselves change. An analogous concept applies to other planets, a planets celestial poles are the points in the sky where the projection of the axis of rotation intersects the celestial sphere. These points vary because different planets axes are oriented differently, Celestial bodies other than Earth also have similarly defined celestial poles. The north celestial pole currently is within a degree of the bright star Polaris, Polaris can, of course, only be seen from locations in the northern hemisphere. Polaris is near the pole for only a small fraction of the 25. It will remain a good approximation for about 1,000 years, to find Polaris, face north and locate the Big Dipper and Little Dipper asterisms. Looking at the cup part of the Big Dipper, imagine that the two stars at the edge of the cup form a line pointing upward out of the cup. This line points directly at the star at the tip of the Little Dippers handle and that star is Polaris, the North Star. The south celestial pole is only from the southern hemisphere. It lies in the dim constellation Octans, the Octant, sigma Octantis is identified as the south pole star, over a degree away from the pole, but with a magnitude of 5.5 it is barely visible on a clear night. The south celestial pole can be located from the Southern Cross, draw an imaginary line from γ Crucis to α Crucis—the two stars at the extreme ends of the long axis of the cross—and follow this line through the sky. This point is 5 or 6 degrees from the celestial pole

27.
Declination
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In astronomy, declination is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declinations angle is measured north or south of the celestial equator, the root of the word declination means a bending away or a bending down. It comes from the root as the words incline and recline. Declination in astronomy is comparable to geographic latitude, projected onto the celestial sphere, points north of the celestial equator have positive declinations, while those south have negative declinations. Any units of measure can be used for declination, but it is customarily measured in the degrees, minutes. Declinations with magnitudes greater than 90° do not occur, because the poles are the northernmost and southernmost points of the celestial sphere, the Earths axis rotates slowly westward about the poles of the ecliptic, completing one circuit in about 26,000 years. This effect, known as precession, causes the coordinates of stationary celestial objects to change continuously, therefore, equatorial coordinates are inherently relative to the year of their observation, and astronomers specify them with reference to a particular year, known as an epoch. Coordinates from different epochs must be rotated to match each other. The currently used standard epoch is J2000.0, which is January 1,2000 at 12,00 TT, the prefix J indicates that it is a Julian epoch. Prior to J2000.0, astronomers used the successive Besselian Epochs B1875.0, B1900.0, the declinations of Solar System objects change very rapidly compared to those of stars, due to orbital motion and close proximity. This similarly occurs in the Southern Hemisphere for objects with less than −90° − φ. An extreme example is the star which has a declination near to +90°. Circumpolar stars never dip below the horizon, conversely, there are other stars that never rise above the horizon, as seen from any given point on the Earths surface. Generally, if a star whose declination is δ is circumpolar for some observer, then a star whose declination is −δ never rises above the horizon, as seen by the same observer. Likewise, if a star is circumpolar for an observer at latitude φ, neglecting atmospheric refraction, declination is always 0° at east and west points of the horizon. At the north point, it is 90° − |φ|, and at the south point, from the poles, declination is uniform around the entire horizon, approximately 0°. Non-circumpolar stars are visible only during certain days or seasons of the year, the Suns declination varies with the seasons. As seen from arctic or antarctic latitudes, the Sun is circumpolar near the summer solstice, leading to the phenomenon of it being above the horizon at midnight

28.
Celestial equator
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The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earths equator. In other words, it is a projection of the terrestrial equator out into space, as a result of the Earths axial tilt, the celestial equator is inclined by 23. 4° with respect to the ecliptic plane. An observer standing on the Earths equator visualizes the celestial equator as a semicircle passing directly overhead through the zenith, as the observer moves north, the celestial equator tilts towards the opposite horizon. Celestial objects near the equator are visible worldwide, but they culminate the highest in the sky in the tropics. The celestial equator currently passes through these constellations, Celestial bodies other than Earth also have similarly defined celestial equators, Celestial pole Celestial sphere Declination Equatorial coordinate system

29.
Equinox
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An equinox is the moment in which the plane of Earths equator passes through the center of the Sun, which occurs twice each year, around 20 March and 23 September. On an equinox, day and night are of equal duration all over the planet. They are not exactly equal, however, due to the size of the sun. To avoid this ambiguity, the word equilux is sometimes used to mean a day in which the durations of light, see Length of equinoctial day and night for further discussion. The word is derived from the Latin aequinoctium, aequus and nox, the equinoxes are the only times when the solar terminator is perpendicular to the equator. As a result, the northern and southern hemispheres are equally illuminated, the word comes from Latin equi or equal and nox meaning night. In other words, the equinoxes are the times when the subsolar point is on the equator. The subsolar point crosses the equator moving northward at the March equinox, the equinoxes, along with solstices, are directly related to the seasons of the year. In the southern hemisphere, the equinox occurs in September. When Julius Caesar established the Julian calendar in 45 BC, he set 25 March as the date of the spring equinox. Because the Julian year is longer than the tropical year. By 1500 AD, it had drifted backwards to 11 March and this drift induced Pope Gregory XIII to create a modern Gregorian calendar. However, the leap year intervals in his calendar were not smooth and this causes the equinox to oscillate by about 53 hours around its mean position. This in turn raised the possibility that it could fall on 22 March, the astronomers chose the appropriate number of days to omit so that the equinox would swing from 19 to 21 March but never fall on the 22nd. Vernal equinox and Autumnal equinox, these names are direct derivatives of Latin. The equivalent common language English terms spring equinox and autumn equinox are even more ambiguous, March equinox and September equinox, names referring to the months of the year they occur, with no ambiguity as to which hemisphere is the context. They are still not universal, however, as not all use a solar-based calendar where the equinoxes occur every year in the same month. Northward equinox and southward equinox, names referring to the apparent direction of motion of the Sun

30.
Celestial longitude
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In astronomy, a celestial coordinate system is a system for specifying positions of celestial objects, satellites, planets, stars, galaxies, and so on. Coordinate systems can specify a position in 3-dimensional space, or merely the direction of the object on the celestial sphere, the coordinate systems are implemented in either spherical coordinates or rectangular coordinates. Spherical coordinates, projected on the sphere, are analogous to the geographic coordinate system used on the surface of the Earth. These differ in their choice of fundamental plane, which divides the sphere into two equal hemispheres along a great circle. Rectangular coordinates, in units, are simply the cartesian equivalent of the spherical coordinates, with the same fundamental plane. Each coordinate system is named after its choice of fundamental plane, the following table lists the common coordinate systems in use by the astronomical community. The fundamental plane divides the sphere into two equal hemispheres and defines the baseline for the latitudinal coordinates, similar to the equator in the geographic coordinate system. The poles are located at ±90° from the fundamental plane, the primary direction is the starting point of the longitudinal coordinates. The origin is the distance point, the center of the celestial sphere. The horizontal, or altitude-azimuth, system is based on the position of the observer on Earth, the positioning of a celestial object by the horizontal system varies with time, but is a useful coordinate system for locating and tracking objects for observers on Earth. It is based on the position of relative to an observers ideal horizon. The equatorial coordinate system is centered at Earths center, but fixed relative to the celestial poles, the coordinates are based on the location of stars relative to Earths equator if it were projected out to an infinite distance. The equatorial describes the sky as seen from the solar system, the equatorial system is the normal coordinate system for most professional and many amateur astronomers having an equatorial mount that follows the movement of the sky during the night. Celestial objects are found by adjusting the telescopes or other instruments scales so that they match the equatorial coordinates of the object to observe. There are also subdivisions into mean of date coordinates, which average out or ignore nutation, and true of date, the fundamental plane is the plane of the Earths orbit, called the ecliptic plane. The geocentric ecliptic system was the coordinate system for ancient astronomy and is still useful for computing the apparent motions of the Sun, Moon. The heliocentric ecliptic system describes the orbital movement around the Sun. The system is used for computing the positions of planets and other solar system bodies

31.
Ecliptic
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The ecliptic is the apparent path of the Sun on the celestial sphere, and is the basis for the ecliptic coordinate system. It also refers to the plane of this path, which is coplanar with the orbit of Earth around the Sun, the motions as described above are simplifications. Due to the movement of Earth around the Earth–Moon center of mass, due to further perturbations by the other planets of the Solar System, the Earth–Moon barycenter wobbles slightly around a mean position in a complex fashion. The ecliptic is actually the apparent path of the Sun throughout the course of a year, because Earth takes one year to orbit the Sun, the apparent position of the Sun also takes the same length of time to make a complete circuit of the ecliptic. With slightly more than 365 days in one year, the Sun moves a little less than 1° eastward every day, again, this is a simplification, based on a hypothetical Earth that orbits at uniform speed around the Sun. The actual speed with which Earth orbits the Sun varies slightly during the year, for example, the Sun is north of the celestial equator for about 185 days of each year, and south of it for about 180 days. The variation of orbital speed accounts for part of the equation of time, if the equator is projected outward to the celestial sphere, forming the celestial equator, it crosses the ecliptic at two points known as the equinoxes. The Sun, in its apparent motion along the ecliptic, crosses the equator at these points, one from south to north. The crossing from south to north is known as the equinox, also known as the first point of Aries. The crossing from north to south is the equinox or descending node. Likewise, the ecliptic itself is not fixed, the gravitational perturbations of the other bodies of the Solar System cause a much smaller motion of the plane of Earths orbit, and hence of the ecliptic, known as planetary precession. The combined action of two motions is called general precession, and changes the position of the equinoxes by about 50 arc seconds per year. Once again, this is a simplification, periodic motions of the Moon and apparent periodic motions of the Sun cause short-term small-amplitude periodic oscillations of Earths axis, and hence the celestial equator, known as nutation. Obliquity of the ecliptic is the used by astronomers for the inclination of Earths equator with respect to the ecliptic. It is about 23. 4° and is currently decreasing 0.013 degrees per hundred years due to planetary perturbations, the angular value of the obliquity is found by observation of the motions of Earth and other planets over many years. From 1984, the Jet Propulsion Laboratorys DE series of computer-generated ephemerides took over as the ephemeris of the Astronomical Almanac. Obliquity based on DE200, which analyzed observations from 1911 to 1979, was calculated, jPLs fundamental ephemerides have been continually updated. J. Laskar computed an expression to order T10 good to 0″. 04/1000 years over 10,000 years, all of these expressions are for the mean obliquity, that is, without the nutation of the equator included

32.
Zodiac
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The zodiac is an area of the sky centered upon the ecliptic, the apparent path of the Sun across the celestial sphere over the course of the year. The paths of the Moon and visible planets also remain close to the ecliptic, within the belt of the zodiac, in western astrology and astronomy, the zodiac is divided into twelve signs, each sign occupying 30° of celestial longitude. Because the signs are regular, they do not correspond exactly to the boundaries of the twelve constellations after which they are named, the English word zodiac derives from zōdiacus, the Latinized form of the Ancient Greek zōidiakòs kýklos, meaning circle of little animals. Zōidion is the diminutive of zōion, the name reflects the prominence of animals among the twelve signs. The construction of the zodiac is described in Ptolemys vast 2nd century AD work, the term zodiac may also refer to the region of the celestial sphere encompassing the paths of the planets corresponding to the band of about eight arc degrees above and below the ecliptic. The zodiac of a planet is the band that contains the path of that particular body, e. g. the zodiac of the Moon is the band of five degrees above. By extension, the zodiac of the comets may refer to the band encompassing most short-period comets, the division of the ecliptic into the zodiacal signs originates in Babylonian astronomy during the first half of the 1st millennium BC. The zodiac draws on stars in earlier Babylonian star catalogues, such as the MUL. APIN catalogue, around the end of the 5th century BC, Babylonian astronomers divided the ecliptic into twelve equal signs, by analogy to twelve schematic months of thirty days each. Each sign contained thirty degrees of longitude, thus creating the first known celestial coordinate system. Because the division was made into equal arcs, 30° each, in Babylonian astronomical diaries, a planet position was generally given with respect to a zodiacal sign alone, less often in specific degrees within a sign. When the degrees of longitude were given, they were expressed with reference to the 30° of the zodiacal sign, in astronomical ephemerides, the positions of significant astronomical phenomena were computed in sexagesimal fractions of a degree. For daily ephemerides, the positions of a planet were not as important as the astrologically significant dates when the planet crossed from one zodiacal sign to the next. The Babylonian star catalogs entered Greek astronomy in the 4th century BC, Babylonia or Chaldea in the Hellenistic world came to be so identified with astrology that Chaldean wisdom became among Greeks and Romans the synonym of divination through the planets and stars. Hellenistic astrology derived in part from Babylonian and Egyptian astrology, horoscopic astrology first appeared in Ptolemaic Egypt. The Dendera zodiac, a dating to ca.50 BC, is the first known depiction of the classical zodiac of twelve signs. The earliest extant Greek text using the Babylonian division of the zodiac into 12 signs of 30 equal degrees each is the Anaphoricus of Hypsicles of Alexandria. Particularly important in the development of Western horoscopic astrology was the astrologer and astronomer Ptolemy, whose work Tetrabiblos laid the basis of the Western astrological tradition. Under the Greeks, and Ptolemy in particular, the planets, Houses, Ptolemy lived in the 2nd century AD, three centuries after the discovery of the precession of the equinoxes by Hipparchus around 130 BC

33.
Singapore Botanic Gardens
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The Singapore Botanic Gardens is a 158-year-old tropical garden located at the fringe of Singapores main shopping belt. It is one of three gardens, and the tropical garden, to be honored as a UNESCO World Heritage Site. The Botanic Gardens has been ranked Asias top park attraction since 2013 and it was declared the inaugural Garden of the Year, International Garden Tourism Awards in 2012, and received Michelin’s three-star rating in 2008. The Botanic Gardens was founded at its present site in 1859 by an agri-horticultural society and it played a pivotal role in the regions rubber trade boom in the early twentieth century, when its first scientific director Henry Nicholas Ridley, headed research into the plants cultivation. By perfecting the technique of rubber extraction, still in use today, at its height in the 1920s, the Malayan peninsula cornered half of the global latex production. Aided by the climate, it houses the largest orchid collection of 1,200 species and 2,000 hybrids. Early in the independence, Singapore Botanic Gardens expertise helped to transform the island into a tropical Garden City. In 1981, the climbing orchid, Vanda Miss Joaquim, was chosen as the nations national flower. Singapores orchid diplomacy honors visiting head of states, dignitaries and celebrities, by naming its finest hybrids after them, Singapores botanic gardens is the only one in the world that opens from 5 a. m. to 12 midnight every day of the year. More than 10,000 species of flora is spread over its 82-hectares area, which is stretched vertically, the Botanic Gardens receives about 4.5 million visitors annually. The first Botanical and Experimental Garden in Singapore was established in 1822 on Government Hill at Fort Canning by Sir Stamford Raffles, the founder of modern Singapore and a keen naturalist. The Gardens main task was to evaluate for cultivation crops which were of potential economic importance including those yielding fruits, vegetables, spices and this first Garden closed in 1829. Lawrence Niven was hired as superintendent and landscape designer to turn what were essentially overgrown plantations, the layout of the Gardens as it is today is largely based on Nivens design. The Singapore Agri-horticultural Society ran out of funds, which led to the government taking over the management of the Gardens in 1874. The first rubber seedlings came to the gardens from Kew Gardens in 1877, a naturalist, Henry Nicholas Ridley, or Mad Ridley as he was known, became director of the gardens in 1888 and spearheaded rubber cultivation. Successful in his experiments with rubber planting, Ridley convinced planters across Malaya to adopt his methods, the results were astounding, Malaya became the worlds number one producer and exporter of natural rubber. Another achievement was the pioneering of orchid hybridisation by Professor Eric Holttum and his techniques led to Singapore being one of the worlds top centres of commercial orchid growing. Today, it has the largest collection of plant specimens

34.
Singapore
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Singapore, officially the Republic of Singapore, sometimes referred to as the Lion City or the Little Red Dot, is a sovereign city-state in Southeast Asia. It lies one degree north of the equator, at the tip of peninsular Malaysia. Singapores territory consists of one island along with 62 other islets. Since independence, extensive land reclamation has increased its size by 23%. During the Second World War, Singapore was occupied by Japan, after early years of turbulence, and despite lacking natural resources and a hinterland, the nation developed rapidly as an Asian Tiger economy, based on external trade and its workforce. Singapore is a global commerce, finance and transport hub, the country has also been identified as a tax haven. Singapore ranks 5th internationally and first in Asia on the UN Human Development Index and it is ranked highly in education, healthcare, life expectancy, quality of life, personal safety, and housing, but does not fare well on the Democracy index. Although income inequality is high, 90% of homes are owner-occupied, 38% of Singapores 5.6 million residents are permanent residents and other foreign nationals. There are four languages on the island, Malay, Mandarin, Tamil. English is its language, most Singaporeans are bilingual. Singapore is a multiparty parliamentary republic, with a Westminster system of unicameral parliamentary government. The Peoples Action Party has won every election since self-government in 1959, however, it is unlikely that lions ever lived on the island, Sang Nila Utama, the Srivijayan prince said to have founded and named the island Singapura, perhaps saw a Malayan tiger. There are however other suggestions for the origin of the name, the central island has also been called Pulau Ujong as far back as the third century CE, literally island at the end in Malay. In 1299, according to the Malay Annals, the Kingdom of Singapura was founded on the island by Sang Nila Utama and these Indianized Kingdoms, a term coined by George Cœdès were characterized by surprising resilience, political integrity and administrative stability. In 1613, Portuguese raiders burned down the settlement, which by then was part of the Johor Sultanate. The wider maritime region and much trade was under Dutch control for the following period, in 1824 the entire island, as well as the Temenggong, became a British possession after a further treaty with the Sultan. In 1826, Singapore became part of the Straits Settlements, under the jurisdiction of British India, prior to Raffles arrival, there were only about a thousand people living on the island, mostly indigenous Malays along with a handful of Chinese. By 1860 the population had swelled to over 80,000, many of these early immigrants came to work on the pepper and gambier plantations

35.
Equator
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The Equator usually refers to an imaginary line on the Earths surface equidistant from the North Pole and South Pole, dividing the Earth into the Northern Hemisphere and Southern Hemisphere. The Equator is about 40,075 kilometres long, some 78. 7% lies across water and 21. 3% over land, other planets and astronomical bodies have equators similarly defined. Generally, an equator is the intersection of the surface of a sphere with the plane that is perpendicular to the spheres axis of rotation. The latitude of the Earths equator is by definition 0° of arc, the equator is the only line of latitude which is also a great circle — that is, one whose plane passes through the center of the globe. The plane of Earths equator when projected outwards to the celestial sphere defines the celestial equator, in the cycle of Earths seasons, the plane of the equator passes through the Sun twice per year, at the March and September equinoxes. To an observer on the Earth, the Sun appears to travel North or South over the equator at these times, light rays from the center of the Sun are perpendicular to the surface of the Earth at the point of solar noon on the Equator. Locations on the Equator experience the quickest sunrises and sunsets because the sun moves nearly perpendicular to the horizon for most of the year. The Earth bulges slightly at the Equator, the diameter of the Earth is 12,750 kilometres. Because the Earth spins to the east, spacecraft must also launch to the east to take advantage of this Earth-boost of speed, seasons result from the yearly revolution of the Earth around the Sun and the tilt of the Earths axis relative to the plane of revolution. During the year the northern and southern hemispheres are inclined toward or away from the sun according to Earths position in its orbit, the hemisphere inclined toward the sun receives more sunlight and is in summer, while the other hemisphere receives less sun and is in winter. At the equinoxes, the Earths axis is not tilted toward the sun, instead it is perpendicular to the sun meaning that the day is about 12 hours long, as is the night, across the whole of the Earth. Near the Equator there is distinction between summer, winter, autumn, or spring. The temperatures are usually high year-round—with the exception of high mountains in South America, the temperature at the Equator can plummet during rainstorms. In many tropical regions people identify two seasons, the wet season and the dry season, but many places close to the Equator are on the oceans or rainy throughout the year, the seasons can vary depending on elevation and proximity to an ocean. The Equator lies mostly on the three largest oceans, the Pacific Ocean, the Atlantic Ocean, and the Indian Ocean. The highest point on the Equator is at the elevation of 4,690 metres, at 0°0′0″N 77°59′31″W and this is slightly above the snow line, and is the only place on the Equator where snow lies on the ground. At the Equator the snow line is around 1,000 metres lower than on Mount Everest, the Equator traverses the land of 11 countries, it also passes through two island nations, though without making a landfall in either. Starting at the Prime Meridian and heading eastwards, the Equator passes through, Despite its name, however, its island of Annobón is 155 km south of the Equator, and the rest of the country lies to the north

36.
Armillary sphere
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As such, it differs from a celestial globe, which is a smooth sphere whose principal purpose is to map the constellations. It was invented separately in ancient Greece and ancient China, with use in the Islamic world. With the Earth as center, a sphere is known as Ptolemaic. With the sun as center, it is known as Copernican, the flag of Portugal features an armillary sphere. The armillary sphere is also featured in Portuguese heraldry, associated with the Portuguese discoveries during the Age of Exploration and this section refers to labels in the diagram below. The exterior parts of machine are a compages of brass rings. The equinoctial A, which is divided into 360 degrees for showing the right ascension in degrees. The Arctic Circle E, and the Antarctic Circle F, each 23½ degrees from its respective pole at N and S.5. The equinoctial colure G, passing through the north and south poles of the heaven at N and S, the solstitial colure H, passing through the poles of the heaven, and through the solstitial points Cancer and Capricorn, in the ecliptic. Within these circular rings is a terrestrial globe J, fixed on an axis K. On this axis is fixed the flat celestial meridian L L and this flat meridian is graduated the same way as the brass meridian of the common globe, and its use is much the same. The globe may be turned by hand within this ring, so as to any given meridian upon it. The horizon is divided into 360 degrees all around its outermost edge, within which are the points of the compass, for showing the amplitude of the sun and the moon, both in degrees and points. The celestial meridian L passes through two notches in the north and south points of the horizon, as in a common globe, both here, if the globe be turned round, the horizon and meridian turn with it. At the south pole of the sphere is a circle of 25 hours, fixed to the rings, in the box T are two wheels and two pinions, whose axes come out at V and U, either of which may be turned by the small winch W. If the earthly globe be turned, the hour-index goes round its hour-circle, but if the sphere be turned, and so, by this construction, the machine is equally fitted to show either the real motion of the earth, or the apparent motion of the heaven. — Then turn the winch, and observe when the sun or moon rise and set in the horizon, throughout Chinese history, astronomers have created celestial globes to assist the observation of the stars. The Chinese also used the sphere in aiding calendrical computations and calculations

37.
Conic section
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In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse, the circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, the conic sections of the Euclidean plane have various distinguishing properties. Many of these have used as the basis for a definition of the conic sections. The type of conic is determined by the value of the eccentricity, in analytic geometry, a conic may be defined as a plane algebraic curve of degree 2, that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. This equation may be written in form, and some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be different from one another. By extending the geometry to a projective plane this apparent difference vanishes, further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically. The conic sections have been studied for thousands of years and have provided a source of interesting. A conic is the curve obtained as the intersection of a plane, called the cutting plane and we shall assume that the cone is a right circular cone for the purpose of easy description, but this is not required, any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point and these are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, we assume that conic refers to a non-degenerate conic. There are three types of conics, the ellipse, parabola, and hyperbola, the circle is a special kind of ellipse, although historically it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the cone and plane is a closed curve, if the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola, in this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is often presented as the following definition, a conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L. For 0 < e <1 we obtain an ellipse, for e =1 a parabola, a circle is a limiting case and is not defined by a focus and directrix, in the plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, an ellipse and a hyperbola each have two foci and distinct directrices for each of them

38.
Hyperbola
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In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other, the hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, each branch of the hyperbola has two arms which become straighter further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the point about which each branch reflects to form the other branch. In the case of the curve f =1 / x the asymptotes are the two coordinate axes, hyperbolas share many of the ellipses analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term, many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids, hyperboloids, hyperbolic geometry, hyperbolic functions, and gyrovector spaces. The word hyperbola derives from the Greek ὑπερβολή, meaning over-thrown or excessive, hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube, but were then called sections of obtuse cones. The term hyperbola is believed to have coined by Apollonius of Perga in his definitive work on the conic sections. The rectangle could be applied to the segment, be shorter than the segment or exceed the segment, the midpoint M of the line segment joining the foci is called the center of the hyperbola. The line through the foci is called the major axis and it contains the vertices V1, V2, which have distance a to the center. The distance c of the foci to the center is called the distance or linear eccentricity. The quotient c a is the eccentricity e, C2 is called the director circle of the hyperbola. In order to get the branch of the hyperbola, one has to use the director circle related to F1. This property should not be confused with the definition of a hyperbola with help of a directrix below, for an arbitrary point the distance to the focus is 2 + y 2 and to the second focus 2 + y 2. Hence the point is on the hyperbola if the condition is fulfilled 2 + y 2 −2 + y 2 = ±2 a. Remove the square roots by suitable squarings and use the relation b 2 = c 2 − a 2 to obtain the equation of the hyperbola, the shape parameters a, b are called the semi major axis and semi minor axis or conjugate axis. As opposed to an ellipse, a hyperbola has two vertices

39.
Ellipse
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In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a type of an ellipse having both focal points at the same location. The shape of an ellipse is represented by its eccentricity, which for an ellipse can be any number from 0 to arbitrarily close to, ellipses are the closed type of conic section, a plane curve resulting from the intersection of a cone by a plane. Ellipses have many similarities with the two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder and this ratio is called the eccentricity of the ellipse. Ellipses are common in physics, astronomy and engineering, for example, the orbit of each planet in our solar system is approximately an ellipse with the barycenter of the planet–Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies, the shapes of planets and stars are often well described by ellipsoids. It is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency, a similar effect leads to elliptical polarization of light in optics. The name, ἔλλειψις, was given by Apollonius of Perga in his Conics, in order to omit the special case of a line segment, one presumes 2 a > | F1 F2 |, E =. The midpoint C of the segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis and it contains the vertices V1, V2, which have distance a to the center. The distance c of the foci to the center is called the distance or linear eccentricity. The quotient c a is the eccentricity e, the case F1 = F2 yields a circle and is included. C2 is called the circle of the ellipse. This property should not be confused with the definition of an ellipse with help of a directrix below, for an arbitrary point the distance to the focus is 2 + y 2 and to the second focus 2 + y 2. Hence the point is on the ellipse if the condition is fulfilled 2 + y 2 +2 + y 2 =2 a. The shape parameters a, b are called the major axis. The points V3 =, V4 = are the co-vertices and it follows from the equation that the ellipse is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin

40.
Circle
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A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre, the point equidistant from the points on the circle. Chord, a segment whose endpoints lie on the circle. Circumference, the length of one circuit along the circle, or the distance around the circle and it is a special case of a chord, namely the longest chord, and it is twice the radius. Disc, the region of the bounded by a circle. Lens, the intersection of two discs, passant, a coplanar straight line that does not touch the circle. Radius, a line segment joining the centre of the circle to any point on the circle itself, or the length of such a segment, sector, a region bounded by two radii and an arc lying between the radii. Segment, a region, not containing the centre, bounded by a chord, secant, an extended chord, a coplanar straight line cutting the circle at two points. Semicircle, an arc that extends from one of a diameters endpoints to the other, in non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disc. A half-disc is a case of a segment, namely the largest one. Tangent, a straight line that touches the circle at a single point. The word circle derives from the Greek κίρκος/κύκλος, itself a metathesis of the Homeric Greek κρίκος, the origins of the words circus and circuit are closely related. The circle has been known since before the beginning of recorded history, natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, in mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Some highlights in the history of the circle are,1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as a value of π.300 BCE – Book 3 of Euclids Elements deals with the properties of circles

41.
History of sundials
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A sundial is a device that measures time by using a light spot or shadow cast by the position of the Sun on a reference scale. Both the azimuth and the altitude can be used to create time measuring devices, Sundials have been invented independently in all major cultures and become more accurate and sophisticated as the culture developed. A sundial uses local solar time, before the coming of the railways in the 1830s and 1840s, local time was displayed on a sundial and was used by the government and commerce. A clock and a dial were used together to measure longitude, dials were laid out using straight edges and compasses. In the late nineteenth century sundials became objects of academic interest, the use of logarithms allowed algebraic methods of laying out dials to be employed and studied. No longer utilitarian, sundials remained as popular ornaments, and several popular books promoted that interest-, affordable scientific calculators made the algebraic methods as accessible as the geometric constructions- and the use of computers made dial plate design trivial. The heritage of sundials was recognised and sundial societies were set up worldwide, the earliest sundials known from the archaeological finds are the shadow clocks in ancient Egyptian astronomy and Babylonian astronomy. Ancient analemmatic sundials of the era and their prototype have been discovered on the territory of modern Russia. Much earlier obelisks, once thought to have used also as sundials. Presumably, humans were telling time from shadow-lengths at an earlier date. In roughly 700 BC, the describes a sundial — the dial of Ahaz mentioned in Isaiah 38,8 and 2 Kings 20,9 — which was likely of Egyptian or Babylonian design. Sundials are believed to have existed in China since ancient times, there is an early reference to sundials from 104 BCE in an assembly of calendar experts. The ancient Greeks developed many of the principles and forms of the sundial, Sundials are believed to have been introduced into Greece by Anaximander of Miletus, c.560 BC. According to Herodotus, the Greeks sundials were initially derived from the Babylonian counterparts, the Greeks were well-positioned to develop the science of sundials, having founded the science of geometry, and in particular discovering the conic sections that are traced by a sundial nodus. The mathematician and astronomer Theodosius of Bithynia is said to have invented a sundial that could be used anywhere on Earth. The Romans adopted the Greek sundials, and the first record of a sun-dial in Rome is 293 BC according to Pliny, plautus complained in one of his plays about his day being chopped into pieces by the ubiquitous sundials. Writing in ca.25 BC, the Roman author Vitruvius listed all the types of dials in Book IX of his De Architectura. All of these are believed to be nodus-type sundials, differing mainly in the surface that receives the shadow of the nodus, the Romans built a very large sundial in 10 BC, the Solarium Augusti, which is a classic nodus-based obelisk casting a shadow on a planar pelekinon

42.
Egyptian astronomy
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Egyptian astronomy begins in prehistoric times, in the Predynastic Period. In the 5th millennium BCE, the circles at Nabta Playa may have made use of astronomical alignments. The Egyptian pyramids were aligned towards the pole star. Roman Egypt produced the greatest astronomer of the era, Ptolemy and his works on astronomy, including the Almagest, became the most influential books in the history of Western astronomy. Following the Muslim conquest of Egypt, the region came to be dominated by Arabic culture, the astronomer Ibn Yunus observed the suns position for many years using a large astrolabe, and his observations on eclipses were still used centuries later. In 1006, Ali ibn Ridwan observed the SN1006, a regarded as the brightest stellar event in recorded history. In the 14th century, Najm al-Din al-Misri wrote a treatise describing over 100 different types of scientific and astronomical instruments, Egyptian astronomy begins in prehistoric times. The presence of stone circles at Nabta Playa in Upper Egypt dating from the 5th millennium BCE show the importance of astronomy to the life of ancient Egypt even in the prehistoric period. The constellation system used among the Egyptians also appears to have been essentially of native origin, the precise orientation of the Egyptian pyramids serves as a lasting demonstration of the high degree of technical skill in watching the heavens attained in the 3rd millennium BCE. It has been shown the pyramids were aligned towards the star, which, because of the precession of the equinoxes, was at that time Thuban. The length of the corridor down which sunlight would travel would have limited illumination at other times of the year, Astronomy played a considerable part in religious matters for fixing the dates of festivals and determining the hours of the night. The titles of several books are preserved recording the movements and phases of the sun, moon. The rising of Sirius at the beginning of the inundation was an important point to fix in the yearly calendar. One of the most important Egyptian astronomical texts was the Book of Nut, beginning with the 9th Dynasty, ancient Egyptians produced Diagonal star tables, which were usually painted on the inside surface of wooden coffin lids. This practice continued until the 12th dynasty and these Diagonal star tables or star charts are also known as diagonal star clocks, in the past they have also been known as star calendars, or decanal clocks. These star charts featuring the paintings of Egyptian deities, decans, constellations, according to the texts, in founding or rebuilding temples the north axis was determined by the same apparatus, and we may conclude that it was the usual one for astronomical observations. In careful hands, it might give results of a degree of accuracy. He named it the Egyptian System, and stated that it did not escape the skill of the Egyptians and he must know by heart the Hermetic astrological books, which are four in number

43.
Babylonian astronomy
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Modern knowledge of Sumerian astronomy is indirect, via the earliest Babylonian star catalogues dating from about 1200 BC. The fact that many names appear in Sumerian suggests a continuity reaching into the Early Bronze Age. The history of astronomy in Mesopotamia, and the world, begins with the Sumerians who developed the earliest writing system—known as cuneiform—around 3500–3200 BC, the Sumerians developed a form of astronomy that had an important influence on the sophisticated astronomy of the Babylonians. Astrolatry, which gave planetary gods an important role in Mesopotamian mythology and religion and they also used a sexagesimal place-value number system, which simplified the task of recording very great and very small numbers. The modern practice of dividing a circle into 360 degrees, of 60 minutes each hour, during the 8th and 7th centuries BC, Babylonian astronomers developed a new empirical approach to astronomy. They began studying philosophy dealing with the nature of the universe. This was an important contribution to astronomy and the philosophy of science and this new approach to astronomy was adopted and further developed in Greek and Hellenistic astronomy. Classical Greek and Latin sources frequently use the term Chaldeans for the astronomers of Mesopotamia, nevertheless, the surviving fragments show that, according to the historian A. Old Babylonian astronomy refers to the astronomy that was practiced during and after the First Babylonian Dynasty, the Babylonians were the first to recognize that astronomical phenomena are periodic and apply mathematics to their predictions. Tablets dating back to the Old Babylonian period document the application of mathematics to the variation in the length of daylight over a solar year and it is the earliest evidence that planetary phenomena were recognized as periodic. There are dozens of cuneiform Mesopotamian texts with real observations of eclipses, the Babylonians were the first civilization known to possess a functional theory of the planets. The Babylonian astrologers also laid the foundations of what would eventually become Western astrology and this is largely due to the current fragmentary state of Babylonian planetary theory, and also due to Babylonian astronomy being independent from cosmology at the time. Nevertheless, traces of cosmology can be found in Babylonian literature and their worldview was not exactly geocentric either. In contrast, Babylonian cosmology suggested that the cosmos revolved around circularly with the heavens, the Babylonians and their predecessors, the Sumerians, also believed in a plurality of heavens and earths. Neo-Babylonian astronomy refers to the astronomy developed by Chaldean astronomers during the Neo-Babylonian, Achaemenid, Seleucid, a significant increase in the quality and frequency of Babylonian observations appeared during the reign of Nabonassar. The systematic records of phenomena in Babylonian astronomical diaries that began at this time allowed for the discovery of a repeating 18-year Saros cycle of lunar eclipses. The Greco-Egyptian astronomer Ptolemy later used Nabonassars reign to fix the beginning of an era, the last stages in the development of Babylonian astronomy took place during the time of the Seleucid Empire. In the 3rd century BC, astronomers began to use texts to predict the motions of the planets

44.
Old Testament
–
Its counterpart is the New Testament, the second part of the Christian Bible. The books that comprise the Old Testament canon differ between Christian Churches as well as their order and names. The most common Protestant canon comprises 39 books, the Catholic canon comprises 46 books, the 39 books in common to all the Christian canons corresponds to 24 books of the Tanakh, with some differences of order, and there are some differences in text. The additional number reflects the split of texts in the Christian Bibles into separate books, for example, Kings, Samuel and Chronicles, Ezra–Nehemiah, the books which are part of a Christian Old Testament but which are not part of the Hebrew canon are sometimes described as deuterocanonical. In general, Protestant bibles do not include books in its canon. The Old Testament consists of translations of many books by various authors produced over a period of centuries. The canon formed in stages, first the Pentateuch by around 400 BC, then the Prophets during the Hasmonean dynasty, and finally the remaining books. The Old Testament contains 39 or 46 or more books, divided, very broadly, into the Pentateuch, the books, the wisdom books. For the Orthodox canon, Septuagint titles are provided in parentheses when these differ from those editions, for the Catholic canon, the Douaic titles are provided in parentheses when these differ from those editions. Likewise, the King James Version references some of these books by the spelling when referring to them in the New Testament. The Talmud in Bava Batra 14b gives a different order for the books in Neviim and Ketuvim and this order is also cited in Mishneh Torah Hilchot Sefer Torah 7,15. The order of the books of the Torah is universal through all denominations of Judaism and they are present in a few historic Protestant versions, the German Luther Bible included such books, as did the English 1611 King James Version. Empty table cells indicate that a book is absent from that canon, several of the books in the Eastern Orthodox canon are also found in the appendix to the Latin Vulgate, formerly the official Bible of the Roman Catholic Church. The books of Joshua, Judges, Samuel and Kings follow, there is a broad consensus among scholars that these originated as a single work during the Babylonian exile of the 6th century BC. The two Books of Chronicles cover much the material as the Pentateuch and Deuteronomistic history and probably date from the 4th century BC. Chronicles, and Ezra–Nehemiah, were finished during the 3rd century BC. Catholic and Orthodox Old Testaments contain two to four Books of Maccabees, written in the 2nd and 1st centuries BC and these history books make up around half the total content of the Old Testament. God is consistently depicted as the one who created or put into order the world, the Old Testament stresses the special relationship between God and his chosen people, Israel, but includes instructions for proselytes as well

45.
Canonical sundial
–
A tide dial, also known as a mass or scratch dial, is a sundial marked with the canonical hours rather than or in addition to the standard hours of daylight. Such sundials were particularly common between the 7th and 14th centuries in Europe, at which point they began to be replaced by mechanical clocks, there are more than 3,000 surviving tide dials in England and at least 1,500 in France. The name tide dial preserves the Old English term tīd, used for hours and canonical hours prior to the Norman Conquest of England, the actual Old English name for sundials was dægmæl or day-marker. Jews long recited prayers at fixed times of day, psalm 119 in particular mentions praising God seven times a day, and the apostles Peter and John are mentioned attending afternoon prayers. The need for monastic communities and others to organize their times of prayer prompted the establishment of tide dials built into the walls of churches. They began to be used in England in the late 7th century and spread from there across continental Europe through copies of Bedes works and by the Saxon, within England, tide dials fell out of favor after the Norman Conquest. By the 13th century, some tide dials—like that at Strasbourg Cathedral—were constructed as independent statues rather than built into the walls of the churches. With Christendom confined to the Northern Hemisphere, the dials were often carved vertically onto the south side of the church chancel at eye level near the priests door. In an abbey or large monastery, dials were carefully carved into the stone walls, some tide dials have a stone gnomon, but many have a circular hole which is used to hold a more easily replaced or adjusted wooden gnomon. These gnomons were perpendicular to the wall and cast a shadow upon the dial, most dials have supplementary lines marking the other 8 daytime hours, but are characterized by their noting the canonical hours particularly. The lines for the canonical hours may be longer or marked with a dot or cross, Dials often have holes along the circumference of their semicircle. The oldest surviving English tide dial is on the 7th- or 8th-century Bewcastle Cross in the graveyard of St Cuthberts in Bewcastle. It is carved on the face of a Celtic cross at some height from the ground and is divided by five principal lines into four tides. Two of these lines, those for 9 am and noon, are crossed at the point, the four spaces are further subdivided so as to give the twelve daylight hours of the Romans. On one side of the dial, there is a line which touches the semicircular border at the second afternoon hour. This may be an accident, but the kind of line is found on the dial in the crypt of Bamburgh Church. The sundial may have used for calculating the date of the spring equinox. Nendrum Monastery in Northern Ireland, supposedly founded in the 5th century by St Machaoi, the 9th-century tide dial gives the name of its sculptor and a priest

Aldeburgh
–
Aldeburgh /ˈɔːlbrə/ is a coastal town in the English county of Suffolk. It remains an artistic and literary centre with an annual Poetry Festival and several festivals as well as other cultural events. It is a former Tudor port and was granted Borough status in 1529 by Henry VIII and its historic buildings include a 16th-century moot hall and a Nap

1.
The Moot Hall

2.
Aldeburgh is the bottom-right settlement depicted in this 1588 map

3.
Elizabeth Garrett Anderson, Mayor of Aldeburgh, 1908

4.
The sundial of the Moot Hall.

Boise, Idaho
–
Boise is the capital and most populous city of the U. S. state of Idaho, as well as the county seat of Ada County. Located on the Boise River in southwestern Idaho, the population of Boise at the 2010 Census was 205,671 and its estimated population in 2013 was 214,237. The Boise-Nampa metropolitan area, also known as the Treasure Valley, includes f

1.
Boise

3.
Downtown Boise in Fall 2013

4.
Floating the Boise River

Time of day
–
An hour is a unit of time conventionally reckoned as 1⁄24 of a day and scientifically reckoned as 3, 599–3,601 seconds, depending on conditions. The seasonal, temporal, or unequal hour was established in the ancient Near East as 1⁄12 of the night or daytime, such hours varied by season, latitude, and weather. It was subsequently divided into 60 m

1.
Key concepts

2.
Midnight on a 24-hour digital clock

Position of the Sun
–
The position of the Sun in the sky is a function of both time and the geographic coordinates of the observer on the surface of the Earth. As the Earth moves around the Sun during the course of the year, the Earths rotation about its axis causes the fixed stars to move in the sky in a way that depends on the observers geographic latitude. The time w

1.
The Sun seen from Lamlash (55° 31′ 47.43″ N, 5° 05′ 59.77″ W) on 3 January 2010, at 8:53 a.m., local time

Sky
–
The sky is everything that lies above the surface of the Earth, including the atmosphere and outer space. In the field of astronomy, the sky is called the celestial sphere. This is viewed from Earths surface as a dome where the Sun, stars, planets. The celestial sphere is divided into regions called constellations. Usually, the sky is used informal

1.
Crepuscular rays of light shining through clouds in the sky

2.
The Crescent Moon remains visible just moments before Sunrise

3.
Aurora borealis over Bear Lake, Alaska

4.
The Milky Way can be seen as a large band across the night sky, and is distorted into an arch in this 360° panorama.

Gnomon
–
A gnomon is the part of a sundial that casts a shadow. The term has come to be used for a variety of purposes in mathematics, a painted stick from 2300 BCE found in China is the oldest known gnomon. Anaximander is credited with introducing this Babylonian instrument to the Greeks, oenopides used the phrase drawn gnomon-wise to describe a line drawn

1.
The gnomon is the triangular blade in this sundial

2.
Gnomon situated on the wall of a building facing Tiradentes Square, Curitiba, Brazil

Shadow
–
A shadow is a dark area where light from a light source is blocked by an opaque object. It occupies all of the volume behind an object with light in front of it. The cross section of a shadow is a silhouette, or a reverse projection of the object blocking the light. A point source of light casts only a shadow, called an umbra. For a non-point or ex

1.
Shadows of visitors to the Eiffel Tower, viewed from the first platform

2.
Park fence shadow is distorted by an uneven snow surface

3.
Three moons and their shadows parade across Jupiter.

4.
Steam phase eruption of Castle Geyser in Yellowstone National Park casts a shadow on its own steam. Crepuscular rays are also visible.

Sun
–
The Sun is the star at the center of the Solar System. It is a perfect sphere of hot plasma, with internal convective motion that generates a magnetic field via a dynamo process. It is by far the most important source of energy for life on Earth. Its diameter is about 109 times that of Earth, and its mass is about 330,000 times that of Earth, accou

1.
The Sun in visible wavelength with filtered white light on 8 July 2014. Characteristic limb darkening and numerous sunspots are visible.

2.
During a total solar eclipse, the solar corona can be seen with the naked eye, during the brief period of totality.

3.
Taken by Hinode 's Solar Optical Telescope on 12 January 2007, this image of the Sun reveals the filamentary nature of the plasma connecting regions of different magnetic polarity.

4.
Visible light photograph of sunspot, 13 December 2006

Earth's rotation
–
Earths rotation is the rotation of the planet Earth around its own axis. The Earth rotates from the west towards east, as viewed from North Star or polestar Polaris, the Earth turns counter-clockwise. The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earths axis of r

1.
Northern night sky over the Nepal Himalayas, showing the paths of the stars as the earth rotates.

2.
An animation showing the rotation of the Earth around its own axis.

Latitude
–
In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earths surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles, lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the

1.
A graticule on the Earth as a sphere or an ellipsoid. The lines from pole to pole are lines of constant longitude, or meridians. The circles parallel to the equator are lines of constant latitude, or parallels. The graticule determines the latitude and longitude of points on the surface. In this example meridians are spaced at 6° intervals and parallels at 4° intervals.

Horizontal coordinate system
–
The horizontal coordinate system is a celestial coordinate system that uses the observers local horizon as the fundamental plane. It is expressed in terms of angle and azimuth. This coordinate system divides the sky into the upper hemisphere where objects are visible, the great circle separating the hemispheres is called the celestial horizon. The

1.
Azimuth is measured from the north point (sometimes from the south point) of the horizon around to the east; altitude is the angle above the horizon.

Azimuth
–
An azimuth is an angular measurement in a spherical coordinate system. An example is the position of a star in the sky, the star is the point of interest, the reference plane is the horizon or the surface of the sea, and the reference vector points north. The azimuth is the angle between the vector and the perpendicular projection of the star down

1.
A standard Brunton Geo compass, used commonly by geologists and surveyors to measure azimuth

2.
The azimuth is the angle formed between a reference direction (North) and a line from the observer to a point of interest projected on the same plane as the reference direction orthogonal to the zenith

Cylindrical lens
–
A cylindrical lens is a lens which focuses light into a line instead of a point, as a spherical lens would. The lens compresses the image in the perpendicular to this line. In a light microscope, a cylindrical lens is placed in front of the illumination objective to create the light sheet used for imaging. Lenticular lens Jacobs, Donald H. Fundamen

1.
Cylindrical lenses.

Plane (mathematics)
–
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the analogue of a point, a line. When working exclusively in two-dimensional Euclidean space, the article is used, so. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a space, or in other words

1.
Vector description of a plane

2.
Two intersecting planes in three-dimensional space

Sphere
–
A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. This distance r is the radius of the ball, and the point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is twic

Cylinder (geometry)
–
In its simplest form, a cylinder is the surface formed by the points at a fixed distance from a given straight line called the axis of the cylinder. It is one of the most basic curvilinear geometric shapes, commonly the word cylinder is understood to refer to a finite section of a right circular cylinder having a finite height with circular ends pe

1.
Tycho Brahe Planetarium building, Copenhagen, its roof being an example of a cylindric section

2.
A right circular cylinder with radius r and height h.

3.
In projective geometry, a cylinder is simply a cone whose apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.

Cone (geometry)
–
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex. A cone is formed by a set of segments, half-lines, or lines connecting a common point. If the enclosed points are included in the base, the cone is a solid object, otherwise it is a two-dimensional object in three-dimensional sp

1.
In projective geometry, a cylinder is simply a cone whose apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.

2.
A right circular cone and an oblique circular cone

Analemmatic sundial
–
Analemmatic sundials are a type of horizontal sundial that has a vertical gnomon and hour markers positioned in an elliptical pattern. The gnomon is not fixed and must change position daily to indicate time of day. Hence there are no lines on the dial and the time of day is read only on the ellipse. As with most sundials, analemmatic sundials mark

1.
Analemmatic sundial at Saint-Etienne, France in which the user's head forms the gnomon of the dial

2.
Analemmatic sundial on a meridian line in the garden of the Herkenrode Abbey in Hasselt (Flanders in Belgium)

Solar time
–
Solar time is a calculation of the passage of time based on the position of the Sun in the sky. The fundamental unit of time is the day. Two types of time are apparent solar time and mean solar time. Fix a tall pole vertically in the ground, at some instant on any day the shadow will point exactly north or south. That instant is local apparent noon

1.
Sun and Moon, Nuremberg Chronicle, 1493

3.
Key concepts

4.
Internal structure

Equation of time
–
The equation of time describes the discrepancy between two kinds of solar time. The word equation is used in the sense of reconcile a difference. The two times that differ are the apparent solar time, which tracks the motion of the sun, and mean solar time. Apparent solar time can be obtained by measurement of the current position of the Sun, Mean

1.
This graph uses the opposite sign to the one above it. There is no universally followed convention for the sign of the equation of time.

2.
The equation of time — above the axis a sundial will appear fast relative to a clock showing local mean time, and below the axis a sundial will appear slow.

3.
A sundial made in 1812, by Whitehurst & Son with a circular scale showing the equation of time correction. This is now on display in the Derby Museum.

Greenwich
–
Greenwich is an early-established district of todays London, England, centred 5.5 miles east south-east of Charing Cross. The town lends its name to the Royal Borough of Greenwich, Greenwich is generally described as being part of South-east London and sometimes as being part of East London. Greenwich is notable for its history and for giving its n

1.
Royal Observatory, Greenwich

2.
The view from Greenwich Park, with the Queen's House and the wings of the National Maritime Museum in the foreground

3.
Boats at Greenwich at the end of the Great River Race

4.
The Royal Navy Type 45 destroyer HMS Defender moored on the riverfront at Greenwich in 2015

Daylight saving time
–
Daylight saving time is the practice of advancing clocks during summer months by one hour so that evening daylight lasts an hour longer, while sacrificing normal sunrise times. Typically, regions that use Daylight Savings Time adjust clocks forward one hour close to the start of spring, American inventor and politician Benjamin Franklin proposed a

4.
William Willett independently proposed DST in 1907 and advocated it tirelessly.

Aberdeen, Scotland
–
Nicknames include the Granite City, the Grey City and the Silver City with the Golden Sands. During the mid-18th to mid-20th centuries, Aberdeens buildings incorporated locally quarried grey granite, since the discovery of North Sea oil in the 1970s, other nicknames have been the Oil Capital of the World or the Energy Capital of the World. The area

1.
From the top: Part of the Aberdeen skyline, Aberdeen Harbour, and the High Street in Old Aberdeen.

4.
The Town House, Old Aberdeen. Once a separate burgh, Old Aberdeen was incorporated into the city in 1891

Sitka, Alaska
–
As of the 2010 census, the population was 8,881. In terms of area, it is the largest city-borough in the U. S. with a land area of 2,870.3 square miles. Urban Sitka, the part that is thought of as the city of Sitka, is on the west side of Baranof Island. The current name Sitka means People on the Outside of Baranof Island, Sitkas location was origi

1.
View toward Sitka from the Pacific Ocean. Sitka is the only town in Southeast Alaska that faces the Gulf of Alaska head-on.

Celestial sphere
–
In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with Earth. All objects in the sky can be thought of as projected upon the inside surface of the celestial sphere. The celestial sphere is a tool for spherical astronomy, allowing observers to plot positions of objects in the sky when th

1.
Celestial Sphere, 18th century. Brooklyn Museum.

2.
The Earth rotating within a relatively small-diameter Earth-centered celestial sphere. Depicted here are stars (white), the ecliptic (red), and lines of right ascension and declination (green) of the equatorial coordinate system.

3.
Celestial globe by Jost Bürgi (1594)

Celestial pole
–
The north and south celestial poles are the two imaginary points in the sky where the Earths axis of rotation, indefinitely extended, intersects the celestial sphere. The north and south celestial poles appear permanently directly overhead to an observer at the Earths North Pole and South Pole respectively. As the Earth spins on its axis, the two c

1.
Over the course of an evening, stars appear to rotate about the north celestial pole. Polaris, within a degree of the pole, is the single nearly-stationary star just to the right of the center of this image.

2.
The north and south celestial poles and their relation to axis of rotation, plane of orbit and axial tilt.

3.
Celestial South Pole over the Very Large Telescope.

4.
Locating the south celestial pole

Declination
–
In astronomy, declination is one of the two angles that locate a point on the celestial sphere in the equatorial coordinate system, the other being hour angle. Declinations angle is measured north or south of the celestial equator, the root of the word declination means a bending away or a bending down. It comes from the root as the words incline a

1.
Right ascension (blue) and declination (green) as seen from outside the celestial sphere.

Celestial equator
–
The celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earths equator. In other words, it is a projection of the terrestrial equator out into space, as a result of the Earths axial tilt, the celestial equator is inclined by 23. 4° with respect to the ecliptic plane. An observer standing on the Earths equ

1.
The celestial equator is inclined by 23.4° to the ecliptic plane. The image shows the relations between Earth's axial tilt (or obliquity), rotation axis and plane of orbit.

Equinox
–
An equinox is the moment in which the plane of Earths equator passes through the center of the Sun, which occurs twice each year, around 20 March and 23 September. On an equinox, day and night are of equal duration all over the planet. They are not exactly equal, however, due to the size of the sun. To avoid this ambiguity, the word equilux is some

1.
The Earth in its orbit around the Sun causes the Sun to appear on the celestial sphere moving over the ecliptic (red), which is tilted on the Equator (white)

2.
Contour plot of the hours of daylight as a function of latitude and day of the year, showing approximately 12 hours of daylight at all latitudes during the equinoxes

3.
Diagram of the Earth's seasons as seen from the north. Far right: December solstice.

4.
Diagram of the Earth's seasons as seen from the south. Far left: June solstice.

Celestial longitude
–
In astronomy, a celestial coordinate system is a system for specifying positions of celestial objects, satellites, planets, stars, galaxies, and so on. Coordinate systems can specify a position in 3-dimensional space, or merely the direction of the object on the celestial sphere, the coordinate systems are implemented in either spherical coordinate

Ecliptic
–
The ecliptic is the apparent path of the Sun on the celestial sphere, and is the basis for the ecliptic coordinate system. It also refers to the plane of this path, which is coplanar with the orbit of Earth around the Sun, the motions as described above are simplifications. Due to the movement of Earth around the Earth–Moon center of mass, due to f

Zodiac
–
The zodiac is an area of the sky centered upon the ecliptic, the apparent path of the Sun across the celestial sphere over the course of the year. The paths of the Moon and visible planets also remain close to the ecliptic, within the belt of the zodiac, in western astrology and astronomy, the zodiac is divided into twelve signs, each sign occupyin

1.
The Earth in its orbit around the Sun causes the Sun to appear on the celestial sphere moving over the ecliptic (red), which is tilted with respect to the equator (blue-white).

2.
Wheel of the zodiac: This 6th century mosaic pavement in a synagogue incorporates Greek-Byzantine elements, Beit Alpha, Israel.

Singapore Botanic Gardens
–
The Singapore Botanic Gardens is a 158-year-old tropical garden located at the fringe of Singapores main shopping belt. It is one of three gardens, and the tropical garden, to be honored as a UNESCO World Heritage Site. The Botanic Gardens has been ranked Asias top park attraction since 2013 and it was declared the inaugural Garden of the Year, Int

1.
Symphony Lake

2.
Singapore Botanic Gardens logo, Cyrtostachys palm

3.
Music was played at this gazebo, known as the Bandstand, in the Singapore Botanic Gardens in the 1930s

4.
Orchids in the National Orchid Garden

Singapore
–
Singapore, officially the Republic of Singapore, sometimes referred to as the Lion City or the Little Red Dot, is a sovereign city-state in Southeast Asia. It lies one degree north of the equator, at the tip of peninsular Malaysia. Singapores territory consists of one island along with 62 other islets. Since independence, extensive land reclamation

1.
Raffles Square around 1900.

2.
Flag

3.
Victorious Japanese troops marching through Singapore City after the British capitulation at the Battle of Singapore in 1942.

4.
A cheering crowd welcome the return of British forces, 1945

Equator
–
The Equator usually refers to an imaginary line on the Earths surface equidistant from the North Pole and South Pole, dividing the Earth into the Northern Hemisphere and Southern Hemisphere. The Equator is about 40,075 kilometres long, some 78. 7% lies across water and 21. 3% over land, other planets and astronomical bodies have equators similarly

1.
Left: A monument marking the Equator near the town of Pontianak, Indonesia Right: Road sign marking the Equator near Nanyuki, Kenya

3.
The Equator marked as it crosses Ilhéu das Rolas, in São Tomé and Príncipe

Armillary sphere
–
As such, it differs from a celestial globe, which is a smooth sphere whose principal purpose is to map the constellations. It was invented separately in ancient Greece and ancient China, with use in the Islamic world. With the Earth as center, a sphere is known as Ptolemaic. With the sun as center, it is known as Copernican, the flag of Portugal fe

1.
Jost Bürgi and Antonius Eisenhoit: Armillary sphere with astronomical clock, made in 1585 in Kassel, now at Nordiska Museet in Stockholm

2.
The original diagram of Su Song 's book of 1092 showing the inner workings of his clocktower; a mechanically-rotated armillary sphere crowns the top.

3.
Celestial globe from the Qing Dynasty

4.
The spherical astrolabe from medieval Islamic astronomy

Conic section
–
In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse, the circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The

Hyperbola
–
In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other, the hyperbola is one of the three kinds of conic section, formed by the intersection

1.
Hyperbolas in the physical world: three cones of light of different widths and intensities are generated by a (roughly) downwards-pointing halogen lamp and its housing. Each cone of light intersects a nearby vertical wall in a hyperbola.

2.
A hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone. The plane does not have to be parallel to the axis of the cone; the hyperbola will be symmetrical in any case.

3.
Hyperbolas produced by interference of waves

4.
Hyperbolas as declination lines on a sundial

Ellipse
–
In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a type of an ellipse having both focal points at the same location. The shape of an ellipse is represented by its ecc

1.
Drawing an ellipse with two pins, a loop, and a pen

2.
An ellipse obtained as the intersection of a cone with an inclined plane.

Circle
–
A circle is a simple closed shape in Euclidean geometry. The distance between any of the points and the centre is called the radius, a circle is a simple closed curve which divides the plane into two regions, an interior and an exterior. Annulus, the object, the region bounded by two concentric circles. Arc, any connected part of the circle, centre

1.
The compass in this 13th-century manuscript is a symbol of God's act of Creation. Notice also the circular shape of the halo

2.
A circle with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre (O) in magenta.

3.
Circular piece of silk with Mongol images

4.
Circles in an old Arabic astronomical drawing.

History of sundials
–
A sundial is a device that measures time by using a light spot or shadow cast by the position of the Sun on a reference scale. Both the azimuth and the altitude can be used to create time measuring devices, Sundials have been invented independently in all major cultures and become more accurate and sophisticated as the culture developed. A sundial

1.
World's oldest sundial, from Egypt's Valley of the Kings (c. 1500 BC), used to measure work hours.

3.
Old sundial located in the Great Mosque of Kairouan also known as the Mosque of Uqba, in Kairouan, Tunisia.

4.
A Scottish gravestone bearing a sundial. The instrument has often doubled as a memento mori.

Egyptian astronomy
–
Egyptian astronomy begins in prehistoric times, in the Predynastic Period. In the 5th millennium BCE, the circles at Nabta Playa may have made use of astronomical alignments. The Egyptian pyramids were aligned towards the pole star. Roman Egypt produced the greatest astronomer of the era, Ptolemy and his works on astronomy, including the Almagest,

1.
Chart from Senemut's tomb, 18th dynasty

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Nut, Egyptian goddess of the sky, with the star chart in the tomb of Ramses VI

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' Star clock ' method from the tomb of Rameses VI

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An Egyptian 30th-dynasty (Ptolemaic) terracotta astrological disc at the Los Angeles County Museum of Art.

Babylonian astronomy
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Modern knowledge of Sumerian astronomy is indirect, via the earliest Babylonian star catalogues dating from about 1200 BC. The fact that many names appear in Sumerian suggests a continuity reaching into the Early Bronze Age. The history of astronomy in Mesopotamia, and the world, begins with the Sumerians who developed the earliest writing system—k

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Babylonian tablet recording Halley's comet in 164 BC.

Old Testament
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Its counterpart is the New Testament, the second part of the Christian Bible. The books that comprise the Old Testament canon differ between Christian Churches as well as their order and names. The most common Protestant canon comprises 39 books, the Catholic canon comprises 46 books, the 39 books in common to all the Christian canons corresponds t

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Jesus

Canonical sundial
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A tide dial, also known as a mass or scratch dial, is a sundial marked with the canonical hours rather than or in addition to the standard hours of daylight. Such sundials were particularly common between the 7th and 14th centuries in Europe, at which point they began to be replaced by mechanical clocks, there are more than 3,000 surviving tide dia

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A canonical sundial at the Notre-Dame church in Uzeste (Gironde, France)

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Yinxu, ruins of an ancient palace dating from the Shang Dynasty (14th century BCE)

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Flag

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Some of the thousands of life-size Terracotta Warriors of the Qin Dynasty, c. 210 BCE

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The Great Wall of China was built by several dynasties over two thousand years to protect the sedentary agricultural regions of the Chinese interior from incursions by nomadic pastoralists of the northern steppes.